{"id":114291,"date":"2026-10-08T22:08:58","date_gmt":"2026-10-08T14:08:58","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/114291.html"},"modified":"2026-10-08T22:08:58","modified_gmt":"2026-10-08T14:08:58","slug":"%e4%b8%89%e7%bb%b4cad%e5%85%b3%e9%94%ae%e6%8a%80%e6%9c%af%e6%8e%a2%e8%ae%a8%ef%bc%88%e5%8d%81%e4%b8%89%ef%bc%89-%e6%8b%93%e6%89%91%e5%ae%9e%e4%bd%93%e9%9d%a2","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/114291.html","title":{"rendered":"\u4e09\u7ef4CAD\u5173\u952e\u6280\u672f\u63a2\u8ba8\uff08\u5341\u4e09\uff09\u2014\u2014\u62d3\u6251\u5b9e\u4f53\u9762"},"content":{"rendered":"<h2>\u7b2c13\u7ae0 \u62d3\u6251\u5b9e\u4f53\u9762<\/h2>\n<p>\u7ae0\u8282\u5b9a\u4f4d&#xff1a;B-Rep \u62d3\u6251\u5c42\u6b21\u4e2d\u8fde\u63a5\u73af\u4e0e\u58f3\u7684\u5173\u952e\u4e8c\u7ef4\u5b9e\u4f53<br \/>\n\u5173\u952e\u8bcd&#xff1a;\u62d3\u6251\u9762&#xff1b;B-Rep&#xff1b;\u53c2\u6570\u57df&#xff1b;p-curve&#xff1b;\u7b2c\u4e00\/\u7b2c\u4e8c\u57fa\u672c\u5f62\u5f0f&#xff1b;\u6cd5\u5411\u5b9a\u5411&#xff1b;\u81ea\u4ea4\u4fee\u590d<\/p>\n<hr \/>\n<h3>\u6458\u8981<\/h3>\n<p>\u9762&#xff08;Face&#xff09;\u662f B-Rep \u62d3\u6251\u5c42\u6b21\u4e2d\u7684\u4e8c\u7ef4\u5b9e\u4f53&#xff0c;\u5173\u8054\u4e00\u4e2a\u66f2\u9762\u4f5c\u4e3a\u51e0\u4f55\u652f\u6491&#xff0c;\u7531\u5916\u73af\u4e0e\u5185\u73af\u754c\u5b9a\u5176\u5728\u53c2\u6570\u57df\u4e0a\u7684\u533a\u57df\u3002\u9762\u662f\u51e0\u4f55\u4e0e\u62d3\u6251\u4ea4\u6c47\u7684\u6838\u5fc3&#xff1a;\u65e2\u627f\u8f7d\u62d3\u6251\u7ed3\u6784&#xff08;\u73af\u8fb9\u754c\u3001\u65b9\u5411\u3001\u90bb\u63a5&#xff09;&#xff0c;\u53c8\u627f\u8f7d\u51e0\u4f55\u4fe1\u606f&#xff08;\u66f2\u9762\u65b9\u7a0b\u3001\u6cd5\u5411\u3001\u66f2\u7387&#xff09;\u3002\u672c\u7ae0\u7cfb\u7edf\u5efa\u7acb\u9762\u7684\u539f\u7406\u3001\u8868\u793a\u3001\u64cd\u4f5c\u4e0e\u5de5\u7a0b\u7ec6\u8282&#xff0c;\u6db5\u76d6\u53c2\u6570\u57df\u4fee\u526a\u3001\u65b9\u5411\u5b9a\u5411\u3001p-curve \u8868\u793a\u3001\u7b2c\u4e00\/\u7b2c\u4e8c\u57fa\u672c\u5f62\u5f0f\u3001\u9762\u79ef\u79ef\u5206\u3001\u70b9\u5728\u9762\u5185\u5224\u5b9a\u3001\u9762\u5230\u70b9\u8ddd\u79bb\u3001\u5e03\u5c14\u5207\u5206\u7b49\u5173\u952e\u5185\u5bb9&#xff0c;\u5e76\u7ed9\u51fa\u4e3b\u6d41\u5185\u6838&#xff08;OCCT\/Parasolid\/CGM\/ACIS\/Granite&#xff09;\u7684\u5b9e\u73b0\u5bf9\u6bd4\u4e0e\u5de5\u7a0b\u8c03\u4f18\u5efa\u8bae\u3002\u672c\u7ae0\u662f\u7406\u89e3 B-Rep \u51e0\u4f55\u8868\u8fbe\u7684\u6838\u5fc3\u4e00\u7ae0&#xff0c;\u4e3a\u7b2c 14 \u7ae0\u58f3\u4e0e\u4f53\u94fa\u57ab\u3002<\/p>\n<hr \/>\n<h3>13.1 \u6982\u8ff0\u4e0e\u80cc\u666f<\/h3>\n<p>\u5728 B-Rep \u62d3\u6251\u5c42\u6b21\u4e2d&#xff0c;\u9762&#xff08;Face&#xff09;\u662f\u8fde\u63a5\u73af\u4e0e\u58f3\u7684\u5173\u952e\u4e8c\u7ef4\u5b9e\u4f53&#xff0c;\u4e5f\u662f\u51e0\u4f55\u9020\u578b\u4e2d\u5de5\u7a0b\u5e08\u6700\u76f4\u63a5\u611f\u77e5\u7684\u5b9e\u4f53\u2014\u2014\u4e00\u4e2a\u96f6\u4ef6\u7684\u6bcf\u4e00\u4e2a\u5e73\u6574\u8868\u9762\u3001\u6bcf\u4e00\u4e2a\u5706\u67f1\u9762\u3001\u6bcf\u4e00\u4e2a\u81ea\u7531\u66f2\u9762\u90fd\u662f B-Rep \u4e2d\u7684\u4e00\u4e2a\u9762\u3002\u9762\u5173\u8054\u4e00\u4e2a\u66f2\u9762\u4f5c\u4e3a\u51e0\u4f55\u652f\u6491&#xff0c;\u5e76\u7531\u4e00\u4e2a\u5916\u73af&#xff08;\u5b9a\u4e49\u4e3b\u4f53\u8fb9\u754c&#xff09;\u548c\u96f6\u6216\u591a\u4e2a\u5185\u73af&#xff08;\u5b9a\u4e49\u5b54&#xff09;\u754c\u5b9a\u5176\u5728\u66f2\u9762\u4e0a\u7684\u533a\u57df\u3002<\/p>\n<p>\u9762\u662f B-Rep \u4e2d\u51e0\u4f55\u4e0e\u62d3\u6251\u4ea4\u6c47\u7684\u6838\u5fc3&#xff1a;<\/p>\n<ul>\n<li>\u62d3\u6251\u7ed3\u6784&#xff1a;\u73af\u8fb9\u754c\u3001\u65b9\u5411\u3001\u90bb\u63a5&#xff1b;<\/li>\n<li>\u51e0\u4f55\u4fe1\u606f&#xff1a;\u66f2\u9762\u65b9\u7a0b\u3001\u6cd5\u5411\u3001\u66f2\u7387\u3002<\/li>\n<\/ul>\n<p>\u51e0\u4e4e\u6240\u6709\u51e0\u4f55\u64cd\u4f5c\u90fd\u56f4\u7ed5\u9762\u5c55\u5f00&#xff1a;<\/p>\n<table>\n<tr>\u64cd\u4f5c\u9762\u5bf9\u5e94\u7684\u89d2\u8272<\/tr>\n<tbody>\n<tr>\n<td>\u6e32\u67d3<\/td>\n<td>\u753b\u9762&#xff08;\u79bb\u6563\u4e3a\u4e09\u89d2\u7f51\u683c&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>\u5e03\u5c14<\/td>\n<td>\u5207\u5206\u9762&#xff08;\u6c42\u4ea4\u4ea7\u751f\u65b0\u8fb9&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>\u7f51\u683c\u5316<\/td>\n<td>\u79bb\u6563\u9762&#xff08;FEA \u7f51\u683c&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>\u8d28\u91cf\u8ba1\u7b97<\/td>\n<td>\u79ef\u9762\u79ef&#xff08;\u9762\u79ef\u5206&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>\u5de5\u7a0b\u56fe<\/td>\n<td>\u6295\u5f71\u9762&#xff08;HLR \u6d88\u9690&#xff09;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u9762\u7684\u8868\u793a\u4e0e\u5904\u7406\u8d28\u91cf\u76f4\u63a5\u51b3\u5b9a\u5185\u6838\u7684\u51e0\u4f55\u8868\u8fbe\u80fd\u529b\u4e0e\u9c81\u68d2\u6027\u3002<\/p>\n<h4>13.1.1 \u9762\u7684\u5de5\u7a0b\u5b9e\u73b0\u516b\u5927\u8981\u7d20<\/h4>\n<p>\u9762\u7684\u5de5\u7a0b\u5b9e\u73b0\u662f B-Rep \u4e2d\u6700\u4e30\u5bcc\u7684&#xff1a;<\/p>\n<li>\u66f2\u9762\u51e0\u4f55&#xff1a;\u5e73\u9762\u3001\u5706\u67f1\u3001\u5706\u9525\u3001\u7403\u3001\u73af\u9762\u7b49\u89e3\u6790\u66f2\u9762&#xff0c;\u4ee5\u53ca NURBS\u3001\u8d1d\u585e\u5c14\u3001\u4fee\u526a\u66f2\u9762\u7b49&#xff0c;\u9700\u652f\u6301\u6c42\u503c\u3001\u6c42\u5bfc\u3001\u6c42\u9006\u3001\u66f2\u7387&#xff1b;<\/li>\n<li>\u53c2\u6570\u57df&#xff1a;\u66f2\u9762 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">S(u,v)S(u,v)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u5728\u53c2\u6570\u57df <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">DD<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><\/span><\/span><\/span><\/span>&#xff08;\u7531\u73af\u5b9a\u4e49&#xff09;\u4e0a\u7684\u90e8\u5206&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">DD<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><\/span><\/span><\/span><\/span> \u662f 2D \u53c2\u6570\u57df\u4e0a\u7684\u4fee\u526a\u533a\u57df&#xff1b;<\/li>\n<li>\u65b9\u5411&#xff1a;\u6cd5\u5411\u662f\u5426\u4e0e\u66f2\u9762\u81ea\u7136\u6cd5\u5411\u4e00\u81f4&#xff0c;\u7531 orientation \u51b3\u5b9a&#xff1b;<\/li>\n<li>\u5bb9\u5dee&#xff1a;\u9762\u5230\u8fb9\u8ddd\u79bb\u5728\u5bb9\u5dee\u5185\u5373\u8d34\u5408&#xff1b;<\/li>\n<li>\u53ef\u4fee\u526a&#xff1a;\u53c2\u6570\u57df <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">DD<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><\/span><\/span><\/span><\/span> \u7531\u5916\u73af <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\u2212-<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6667em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">\u2212<\/span><\/span><\/span><\/span><\/span> \u5185\u73af\u5b9a\u4e49&#xff0c;\u652f\u6301\u4efb\u610f\u5f62\u72b6\u9762&#xff1b;<\/li>\n<li>\u53ef\u81ea\u4ea4&#xff1a;\u66f2\u9762\u81ea\u4ea4\u9700\u68c0\u6d4b\u4fee\u590d&#xff1b;<\/li>\n<li>\u5e03\u5c14\u5207\u5206&#xff1a;\u6c42\u4ea4\u4ea7\u751f\u65b0\u8fb9&#xff0c;\u65b0\u8fb9\u628a\u9762\u5207\u5206\u4e3a\u7247\u6bb5&#xff1b;<\/li>\n<li>p-curve&#xff1a;\u73af\u8fb9\u5728\u53c2\u6570\u57df\u7684\u50cf\u3002<\/li>\n<p>\u672c\u7ae0\u5c06\u7cfb\u7edf\u5efa\u7acb\u9762\u7684\u539f\u7406\u3001\u8868\u793a\u3001\u64cd\u4f5c\u4e0e\u5de5\u7a0b\u7ec6\u8282\u3002<\/p>\n<hr \/>\n<h3>13.2 \u5386\u53f2\u4e0e\u53d1\u5c55\u6f14\u8fdb<\/h3>\n<p>\u9762\u4f5c\u4e3a\u62d3\u6251\u5b9e\u4f53\u4e0e B-Rep \u540c\u751f\u3002<\/p>\n<table>\n<tr>\u65f6\u671f\u8fdb\u5c55<\/tr>\n<tbody>\n<tr>\n<td>1970s<\/td>\n<td>Braid BUILD-1 \u4e2d\u9762\u5173\u8054\u66f2\u9762\u4e0e\u73af&#xff1b;\u65e9\u671f\u9762\u4ec5\u652f\u6301\u89e3\u6790\u66f2\u9762&#xff08;\u5e73\u9762\u3001\u5706\u67f1\u7b49&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>1980\u201390s<\/td>\n<td>NURBS \u6210\u719f&#xff0c;\u9762\u53ef\u5173\u8054 NURBS \u66f2\u9762&#xff0c;\u8868\u8fbe\u81ea\u7531\u66f2\u9762&#xff1b;\u4fee\u526a\u66f2\u9762\u968f NURBS \u666e\u53ca<\/td>\n<\/tr>\n<tr>\n<td>1990s<\/td>\n<td>\u9762\u7684\u65b9\u5411\u4e0e orientation \u5728\u534a\u8fb9\u7ed3\u6784\u4e2d\u901a\u8fc7\u6cd5\u5411\u7ea6\u5b9a\u5904\u7406&#xff1b;OCCT \u4e2d\u901a\u8fc7 TopoDS_Face \u7684 Orientation&#xff08;FORWARD\/REVERSED&#xff09;\u8868\u793a<\/td>\n<\/tr>\n<tr>\n<td>1990s\u20132000s<\/td>\n<td>\u9762\u7684\u81ea\u4ea4\u68c0\u6d4b\u4e0e\u4fee\u590d\u968f Shape Healing \u6210\u719f&#xff1b;OCCT \u7684 ShapeFix_Face\u3001ShapeUpgrade \u4fee\u590d\u9762\u81ea\u4ea4\u3001\u9000\u5316\u3001\u9519\u5411<\/td>\n<\/tr>\n<tr>\n<td>2000s<\/td>\n<td>\u9762\u7684\u53c2\u6570\u57df\u8868\u793a&#xff08;Geom2d \u66f2\u7ebf p-curve&#xff09;\u968f\u4fee\u526a\u66f2\u9762\u6210\u719f&#xff0c;\u652f\u6301\u9762\u4e0a\u64cd\u4f5c\u5728 2D \u53c2\u6570\u57df\u8fdb\u884c<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u81f3\u4eca&#xff0c;\u9762\u7684\u8868\u793a\u4e0e\u5904\u7406\u5df2\u6807\u51c6\u5316&#xff0c;\u5404\u5185\u6838\u5dee\u5f02\u5728\u66f2\u9762\u7c7b\u578b\u652f\u6301\u4e0e\u5bb9\u5dee\u6a21\u578b\u3002<\/p>\n<hr \/>\n<h3>13.3 \u6838\u5fc3\u6982\u5ff5\u4e0e\u5b9a\u4e49<\/h3>\n<h4>13.3.1 \u9762&#xff08;Face&#xff09;<\/h4>\n<p>B-Rep \u62d3\u6251\u5c42\u6b21\u7684\u4e8c\u7ef4\u5b9e\u4f53&#xff0c;\u5173\u8054\u4e00\u4e2a\u66f2\u9762&#xff08;\u51e0\u4f55\u652f\u6491&#xff09;&#xff0c;\u7531\u5916\u73af &#043; \u5185\u73af\u754c\u5b9a\u53c2\u6570\u57df\u533a\u57df\u3002<\/p>\n<h4>13.3.2 \u51e0\u4f55\u652f\u6491<\/h4>\n<p>\u9762\u5173\u8054\u4e00\u4e2a\u66f2\u9762&#xff1a;\u5e73\u9762&#xff08;Geom_Plane&#xff09;\u3001\u5706\u67f1&#xff08;Geom_CylindricalSurface&#xff09;\u3001\u5706\u9525\u3001\u7403\u3001\u73af\u9762&#xff08;\u89e3\u6790\u66f2\u9762&#xff09;\u3001NURBS \u66f2\u9762&#xff08;Geom_BSplineSurface&#xff09;\u3001\u8d1d\u585e\u5c14\u66f2\u9762\u3001\u65cb\u8f6c\u66f2\u9762\u3001\u62c9\u4f38\u66f2\u9762\u7b49\u3002<\/p>\n<h4>13.3.3 \u53c2\u6570\u57df\u4e0e\u4fee\u526a<\/h4>\n<p>\u9762\u5173\u8054\u66f2\u9762 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">S(u,v)S(u,v)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u4e0e\u53c2\u6570\u57df <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">DD<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><\/span><\/span><\/span><\/span>\u3002<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">DD<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><\/span><\/span><\/span><\/span> \u7531\u5916\u73af&#xff08;\u4e3b\u4f53&#xff09;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\u2212-<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6667em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">\u2212<\/span><\/span><\/span><\/span><\/span> \u5185\u73af&#xff08;\u5b54&#xff09;\u5b9a\u4e49\u3002\u9762\u662f&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">F&#061;S(D)&#061;{S(u,v):(u,v)\u2208D}<br \/>\nF &#061; S(D) &#061; \\\\{ S(u,v) : (u,v) \\\\in D \\\\}<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">{<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">:<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2208<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u4fee\u526a\u66f2\u9762\u652f\u6301\u4efb\u610f\u5f62\u72b6\u9762\u3002<\/p>\n<h4>13.3.4 \u5916\u73af\u4e0e\u5185\u73af<\/h4>\n<ul>\n<li>\u5916\u73af&#xff1a;\u5b9a\u4e49\u9762\u4e3b\u4f53\u8fb9\u754c&#xff08;\u53c2\u6570\u57df\u5916\u8fb9\u754c&#xff09;&#xff1b;<\/li>\n<li>\u5185\u73af&#xff1a;\u5b9a\u4e49\u5b54&#xff08;\u53c2\u6570\u57df\u5185\u8fb9\u754c&#xff09;\u3002<\/li>\n<\/ul>\n<p>\u73af\u5728\u53c2\u6570\u57df\u4e0a\u662f p-curve\u3002<\/p>\n<h4>13.3.5 \u65b9\u5411&#xff08;Orientation&#xff09;<\/h4>\n<p>FORWARD&#xff08;\u6cd5\u5411\u53d6\u66f2\u9762\u81ea\u7136\u6cd5\u5411&#xff09;\u6216 REVERSED&#xff08;\u53d6\u53cd&#xff09;\u3002\u9762\u6cd5\u5411&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">n(u,v)&#061;orientation\u00d7\u2202S\/\u2202u\u00d7\u2202S\/\u2202v\u2223\u2202S\/\u2202u\u00d7\u2202S\/\u2202v\u2223<br \/>\n\\\\mathbf{n}(u,v) &#061; \\\\text{orientation} \\\\times \\\\frac{\\\\partial S\/\\\\partial u \\\\times \\\\partial S\/\\\\partial v}{\\\\left| \\\\partial S\/\\\\partial u \\\\times \\\\partial S\/\\\\partial v \\\\right|}<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathbf\">n<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7512em;vertical-align: -0.0833em\"><\/span><span class=\"mord text\"><span class=\"mord\">orientation<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.363em;vertical-align: -0.936em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\">\u2223<\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mord\">\/<\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mord\">\/<\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose delimcenter\" style=\"top: 0em\">\u2223<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mord\">\/<\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mord\">\/<\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.936em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h4>13.3.6 \u5bb9\u5dee&#xff08;Tolerance&#xff09;<\/h4>\n<p>\u9762\u5e26\u5bb9\u5dee <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\u03b5\\\\varepsilon<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">\u03b5<\/span><\/span><\/span><\/span><\/span>\u3002\u8fb9\u5230\u9762\u8ddd\u79bb <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\u2264\u03b5\\\\leq \\\\varepsilon<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7719em;vertical-align: -0.136em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">\u03b5<\/span><\/span><\/span><\/span><\/span> \u5373\u8d34\u5408\u3002\u5bb9\u5dee\u9762\u662f\u5bb9\u5dee\u5efa\u6a21\u7684\u57fa\u7840\u3002<\/p>\n<h4>13.3.7 \u81ea\u4ea4<\/h4>\n<p>\u66f2\u9762 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">S(u,v)S(u,v)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u81ea\u4ea4\u2014\u2014\u5b58\u5728 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">(u1,v1)\u2260(u2,v2)(u_1, v_1) \\\\neq (u_2, v_2)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\"><span class=\"mrel\"><span class=\"mord vbox\"><span class=\"thinbox\"><span class=\"rlap\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"inner\"><span class=\"mord\"><span class=\"mrel\">\ue020<\/span><\/span><\/span><span class=\"fix\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace nobreak\"><\/span><span class=\"mrel\">&#061;<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u4f7f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">S(u1,v1)&#061;S(u2,v2)S(u_1, v_1) &#061; S(u_2, v_2)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002\u5408\u6cd5\u9762\u4e0d\u5e94\u81ea\u4ea4&#xff0c;\u81ea\u4ea4\u9700\u4fee\u590d\u3002<\/p>\n<h4>13.3.8 \u53c2\u6570\u8868\u793a\u4e0e p-curve<\/h4>\n<p>\u9762\u4e0a\u70b9 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">S(u,v)S(u,v)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">(u,v)\u2208D(u,v) \\\\in D<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2208<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><\/span><\/span><\/span><\/span>\u3002\u6cd5\u5411\u3001\u66f2\u7387\u5728\u53c2\u6570\u57df\u8ba1\u7b97\u3002p-curve \u662f\u9762\u73af\u8fb9\u5728\u53c2\u6570\u57df\u7684\u50cf&#xff08;Geom2d_Curve&#xff09;&#xff0c;\u4f7f\u9762\u4e0a\u8fb9\u64cd\u4f5c\u5728 2D \u53c2\u6570\u57df\u8fdb\u884c\u3002<\/p>\n<h4>13.3.9 \u5171\u4eab<\/h4>\n<p>\u4e24\u9762\u5171\u4eab\u8fb9\u65f6&#xff0c;\u8fb9\u5bf9\u8c61\u5171\u4eab\u3002\u9762\u672c\u8eab\u88ab\u58f3\u5f15\u7528\u65f6\u5171\u4eab\u3002<\/p>\n<hr \/>\n<h3>13.4 \u6280\u672f\u539f\u7406\u8be6\u89e3<\/h3>\n<p>\u9762\u7684\u5de5\u7a0b\u5b9e\u73b0\u56f4\u7ed5\u201c\u8868\u793a\u3001\u521b\u5efa\u3001\u67e5\u8be2\u3001\u6c42\u503c\u3001\u65b9\u5411\u3001\u4fee\u526a\u3001\u81ea\u4ea4\u201d\u5c55\u5f00\u3002<\/p>\n<h4>13.4.1 \u8868\u793a<\/h4>\n<p>OCCT \u7684 TopoDS_Face \u662f\u8f7b\u91cf\u53e5\u67c4&#xff0c;\u5b9e\u9645\u6570\u636e\u5728 BRep_TFace&#xff08;TShape&#xff09;\u4e2d&#xff1a;<\/p>\n<table>\n<tr>\u5b57\u6bb5\u542b\u4e49<\/tr>\n<tbody>\n<tr>\n<td>surface<\/td>\n<td>\u66f2\u9762&#xff08;Geom_Surface&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>wires<\/td>\n<td>\u73af\u5217\u8868&#xff08;\u5916\u73af &#043; \u5185\u73af&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>tolerance<\/td>\n<td>\u5bb9\u5dee<\/td>\n<\/tr>\n<tr>\n<td>location<\/td>\n<td>\u4f4d\u7f6e\u53d8\u6362<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>TopoDS_Face \u52a0 Orientation\u3002<\/p>\n<h4>13.4.2 \u521b\u5efa<\/h4>\n<table>\n<tr>API\u8bed\u4e49<\/tr>\n<tbody>\n<tr>\n<td>BRepBuilderAPI_MakeFace(surface)<\/td>\n<td>\u4ece\u66f2\u9762\u521b\u5efa&#xff08;\u65e0\u4fee\u526a&#xff0c;\u5168\u53c2\u6570\u57df&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>BRepBuilderAPI_MakeFace(surface, wire)<\/td>\n<td>\u4ece\u66f2\u9762\u4e0e\u5916\u73af\u521b\u5efa<\/td>\n<\/tr>\n<tr>\n<td>BRepBuilderAPI_MakeFace(wire)<\/td>\n<td>\u4ece\u5e73\u9762\u73af\u521b\u5efa\u5e73\u9762\u9762<\/td>\n<\/tr>\n<tr>\n<td>\u4f53\u7d20\u521b\u5efa<\/td>\n<td>\u751f\u6210\u9762&#xff08;\u5982\u7acb\u65b9\u4f53 6 \u4e2a\u5e73\u9762\u9762&#xff09;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>13.4.3 \u67e5\u8be2<\/h4>\n<table>\n<tr>\u67e5\u8be2\u9879API<\/tr>\n<tbody>\n<tr>\n<td>\u66f2\u9762<\/td>\n<td>BRep_Tool::Surface(face) \u8fd4\u56de Geom_Surface<\/td>\n<\/tr>\n<tr>\n<td>\u73af<\/td>\n<td>face.OuterWire\u3001face.Wires&#xff08;\u5916\u73af &#043; \u5185\u73af&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>\u5bb9\u5dee<\/td>\n<td>BRep_Tool::Tolerance(face)<\/td>\n<\/tr>\n<tr>\n<td>\u65b9\u5411<\/td>\n<td>face.Orientation()<\/td>\n<\/tr>\n<tr>\n<td>\u53c2\u6570\u57df<\/td>\n<td>Geom_Surface::Bounds \u8fd4\u56de <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">(u1,u2,v1,v2)(u_1, u_2, v_1, v_2)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u5168\u57df&#xff0c;\u5b9e\u9645\u57df <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">DD<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><\/span><\/span><\/span><\/span> \u7531\u73af\u5b9a\u4e49<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>13.4.4 \u6c42\u503c<\/h4>\n<p>\u9762\u4e0a\u4e00\u70b9 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">S(u,v)S(u,v)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff1a;Geom_Surface::Value(u,v)\u3002\u6cd5\u5411&#xff1a;D1(u,v) \u8fd4\u56de <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\u2202S\/\u2202u,\u2202S\/\u2202v\\\\partial S\/\\\\partial u, \\\\partial S\/\\\\partial v<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mord\">\/<\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mord\">\/<\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u53c9\u79ef\u5f97\u6cd5\u5411\u3002\u66f2\u7387&#xff1a;D2 \u8fd4\u56de\u4e8c\u9636\u5bfc&#xff0c;\u8ba1\u7b97\u9ad8\u65af\u66f2\u7387\u3001\u4e3b\u66f2\u7387\u3002BRepAdaptor_Surface \u5c01\u88c5\u3002<\/p>\n<h4>13.4.5 \u65b9\u5411\u5904\u7406<\/h4>\n<p>\u9762\u6cd5\u5411 &#061; <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">orientation\u00d7(\u2202S\/\u2202u\u00d7\u2202S\/\u2202v)\\\\text{orientation} \\\\times (\\\\partial S\/\\\\partial u \\\\times \\\\partial S\/\\\\partial v)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7512em;vertical-align: -0.0833em\"><\/span><span class=\"mord text\"><span class=\"mord\">orientation<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mord\">\/<\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mord\">\/<\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002FORWARD \u53d6\u81ea\u7136\u6cd5\u5411&#xff0c;REVERSED \u53d6\u53cd\u3002\u65b9\u5411\u51b3\u5b9a\u9762\u5728\u4f53\u4e2d\u671d\u5916\u8fd8\u662f\u671d\u5185&#xff08;\u5916\u58f3\u9762\u6cd5\u5411\u671d\u5916\u3001\u5185\u58f3\u671d\u5185&#xff09;\u3002<\/p>\n<h4>13.4.6 \u4fee\u526a\u4e0e\u53c2\u6570\u57df<\/h4>\n<p>\u9762\u53c2\u6570\u57df <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">DD<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><\/span><\/span><\/span><\/span> \u7531\u5916\u73af <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\u2212-<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6667em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">\u2212<\/span><\/span><\/span><\/span><\/span> \u5185\u73af\u5b9a\u4e49\u3002\u70b9\u5728\u9762\u4e0a&#xff1a;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">S\u22121(p)\u2208DS^{-1}(p) \\\\in D<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">p<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2208<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><\/span><\/span><\/span><\/span>\u3002\u70b9\u5728\u9762\u5185\u5224\u5b9a&#xff1a;\u53c2\u6570 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">(u,v)(u,v)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u5728\u5916\u73af\u5185\u4e14\u6240\u6709\u5185\u73af\u5916\u3002\u9762\u79ef&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">A&#061;\u222cD\u2223\u2202S\u2202u\u00d7\u2202S\u2202v\u2223\u2009du\u2009dv<br \/>\nA &#061; \\\\iint_D \\\\left| \\\\frac{\\\\partial S}{\\\\partial u} \\\\times \\\\frac{\\\\partial S}{\\\\partial v} \\\\right| \\\\, du \\\\, dv<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.4em;vertical-align: -0.95em\"><\/span><span class=\"mop\"><span class=\"mop op-symbol large-op\" style=\"margin-right: 0.4445em;position: relative;top: -0.001em\">\u222c<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"\" style=\"top: -1.7883em;margin-left: -0.4445em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0278em\">D<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9117em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\"><span class=\"mopen\"><span class=\"delimsizing mult\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.45em\"><span class=\"\" style=\"top: -3.45em\"><span class=\"pstrut\" style=\"height: 4.4em\"><\/span><span class=\"\" style=\"width: 0.333em;height: 2.4em\"><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.95em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3714em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\">u<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3714em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose\"><span class=\"delimsizing mult\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.45em\"><span class=\"\" style=\"top: -3.45em\"><span class=\"pstrut\" style=\"height: 4.4em\"><\/span><span class=\"\" style=\"width: 0.333em;height: 2.4em\"><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.95em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h4>13.4.7 p-curve<\/h4>\n<p>\u9762\u73af\u8fb9\u5728\u53c2\u6570\u57df\u7684\u50cf\u3002BRep_Tool::CurveOnSurface(edge, surface) \u67e5\u8be2\u3002p-curve \u4f7f\u9762\u4e0a\u8fb9\u64cd\u4f5c&#xff08;\u6c42\u4ea4\u3001\u5206\u7c7b&#xff09;\u5728 2D \u53c2\u6570\u57df\u8fdb\u884c&#xff0c;\u964d\u7ef4\u63d0\u6548\u3002<\/p>\n<h4>13.4.8 \u81ea\u4ea4\u68c0\u6d4b<\/h4>\n<p>\u66f2\u9762\u81ea\u4ea4\u68c0\u6d4b&#xff1a;\u91c7\u6837\u53c2\u6570\u57df&#xff0c;\u627e <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">S(u1,v1)&#061;S(u2,v2)S(u_1, v_1) &#061; S(u_2, v_2)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u7684 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">(u1,v1)\u2260(u2,v2)(u_1, v_1) \\\\neq (u_2, v_2)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\"><span class=\"mrel\"><span class=\"mord vbox\"><span class=\"thinbox\"><span class=\"rlap\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"inner\"><span class=\"mord\"><span class=\"mrel\">\ue020<\/span><\/span><\/span><span class=\"fix\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace nobreak\"><\/span><span class=\"mrel\">&#061;<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002ShapeFix_Face \u6216 IntTools \u68c0\u6d4b\u3002\u81ea\u4ea4\u9762\u9700\u4fee\u590d&#xff08;\u91cd\u5efa\u6a21\u6216\u62c6\u5206&#xff09;\u3002<\/p>\n<h4>13.4.9 \u5e03\u5c14\u5207\u5206<\/h4>\n<p>\u5e03\u5c14\u6c42\u4ea4\u4ea7\u751f\u65b0\u8fb9&#xff08;\u4ea4\u7ebf&#xff09;&#xff0c;\u65b0\u8fb9\u628a\u9762\u5207\u5206\u4e3a\u201c\u5728\u5bf9\u65b9\u5185\/\u5916\/\u8fb9\u754c\u201d\u7247\u6bb5\u3002\u5207\u5206\u901a\u8fc7 mef&#xff08;make edge face&#xff09;\u52a0\u8fb9\u52a0\u9762\u5b9e\u73b0\u3002<\/p>\n<h4>13.4.10 \u9762\u5230\u70b9\u8ddd\u79bb<\/h4>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">d&#061;min\u2061(u,v)\u2208D\u2223S(u,v)\u2212p\u2223<br \/>\nd &#061; \\\\min_{(u,v) \\\\in D} \\\\left| S(u,v) &#8211; \\\\mathbf{p} \\\\right|<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6944em\"><\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.716em;vertical-align: -0.966em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6679em\"><span class=\"\" style=\"top: -2.309em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">u<\/span><span class=\"mpunct mtight\">,<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose mtight\">)<\/span><span class=\"mrel mtight\">\u2208<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0278em\">D<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"\"><span class=\"mop\">min<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.966em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathbf\">p<\/span><span class=\"mclose delimcenter\" style=\"top: 0em\">\u2223<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u6570\u503c\u4f18\u5316&#xff08;\u725b\u987f&#xff09;\u6216\u91c7\u6837\u3002\u7528\u4e8e\u62fe\u53d6\u3001\u6700\u8fd1\u70b9\u3002<\/p>\n<hr \/>\n<h3>13.5 \u6570\u5b66\u7406\u8bba\u4e0e\u516c\u5f0f\u63a8\u5bfc<\/h3>\n<h4>13.5.1 \u9762\u7684\u51e0\u4f55\u8868\u793a<\/h4>\n<p>\u9762 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">FF<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><\/span><\/span><\/span><\/span> \u5173\u8054\u66f2\u9762 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">S:D\u2192R3S: D \\\\to \\\\mathbb{R}^3<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">:<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2192<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em\"><\/span><span class=\"mord\"><span class=\"mord mathbb\">R<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">(u,v)\u2208D(u,v) \\\\in D<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2208<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><\/span><\/span><\/span><\/span>\u3002\u9762\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">S(D)S(D)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">DD<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><\/span><\/span><\/span><\/span> \u7531\u5916\u73af <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\u2212-<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6667em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">\u2212<\/span><\/span><\/span><\/span><\/span> \u5185\u73af\u5b9a\u4e49\u3002<\/p>\n<h4>13.5.2 \u53c2\u6570\u57df\u4e0e\u4fee\u526a<\/h4>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">D&#061;Douter\u2216(Dinner1\u222aDinner2\u222a\u22ef\u2009)<br \/>\nD &#061; D_{\\\\text{outer}} \\\\setminus (D_{\\\\text{inner1}} \\\\cup D_{\\\\text{inner2}} \\\\cup \\\\cdots)<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2806em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord text mtight\"><span class=\"mord mtight\">outer<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2216<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3175em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord text mtight\"><span class=\"mord mtight\">inner1<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u222a<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3175em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord text mtight\"><span class=\"mord mtight\">inner2<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u222a<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"minner\">\u22ef<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">DouterD_{\\\\text{outer}}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2806em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord text mtight\"><span class=\"mord mtight\">outer<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u5916\u73af\u5185&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">DinnerD_{\\\\text{inner}}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3175em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord text mtight\"><span class=\"mord mtight\">inner<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u5185\u73af\u5185\u3002\u9762 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">&#061;S(D)&#061; S(D)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.3669em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<h4>13.5.3 \u6c42\u503c\u4e0e\u6c42\u5bfc<\/h4>\n<p><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">S(u,v)S(u,v)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u7528 de Boor&#xff08;NURBS&#xff09;\u6216\u89e3\u6790\u3002<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\u2202S\/\u2202u\\\\partial S\/\\\\partial u<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mord\">\/<\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\">u<\/span><\/span><\/span><\/span><\/span>\u3001<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\u2202S\/\u2202v\\\\partial S\/\\\\partial v<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mord\">\/<\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><\/span><\/span><\/span><\/span> \u4e00\u9636\u504f\u5bfc\u3002\u4e8c\u9636\u504f\u5bfc <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\u22022S\/\u2202u2\\\\partial^2 S\/\\\\partial u^2<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mord\">\/<\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord\"><span class=\"mord mathnormal\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3001<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\u22022S\/\u2202u\u2202v\\\\partial^2 S\/\\\\partial u \\\\partial v<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mord\">\/<\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><\/span><\/span><\/span><\/span>\u3001<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\u22022S\/\u2202v2\\\\partial^2 S\/\\\\partial v^2<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mord\">\/<\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<h4>13.5.4 \u6cd5\u5411<\/h4>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">n(u,v)&#061;orientation\u00d7\u2202S\/\u2202u\u00d7\u2202S\/\u2202v\u2223\u2202S\/\u2202u\u00d7\u2202S\/\u2202v\u2223<br \/>\n\\\\mathbf{n}(u,v) &#061; \\\\text{orientation} \\\\times \\\\frac{\\\\partial S\/\\\\partial u \\\\times \\\\partial S\/\\\\partial v}{\\\\left| \\\\partial S\/\\\\partial u \\\\times \\\\partial S\/\\\\partial v \\\\right|}<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathbf\">n<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7512em;vertical-align: -0.0833em\"><\/span><span class=\"mord text\"><span class=\"mord\">orientation<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.363em;vertical-align: -0.936em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\">\u2223<\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mord\">\/<\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mord\">\/<\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose delimcenter\" style=\"top: 0em\">\u2223<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mord\">\/<\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mord\">\/<\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.936em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h4>13.5.5 \u7b2c\u4e00\u57fa\u672c\u5f62\u5f0f&#xff08;\u5ea6\u91cf&#xff09;<\/h4>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">E&#061;\u2202S\u2202u\u22c5\u2202S\u2202u,F&#061;\u2202S\u2202u\u22c5\u2202S\u2202v,G&#061;\u2202S\u2202v\u22c5\u2202S\u2202v<br \/>\nE &#061; \\\\frac{\\\\partial S}{\\\\partial u} \\\\cdot \\\\frac{\\\\partial S}{\\\\partial u}, \\\\quad<br \/>\nF &#061; \\\\frac{\\\\partial S}{\\\\partial u} \\\\cdot \\\\frac{\\\\partial S}{\\\\partial v}, \\\\quad<br \/>\nG &#061; \\\\frac{\\\\partial S}{\\\\partial v} \\\\cdot \\\\frac{\\\\partial S}{\\\\partial v}<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">E<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0574em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3714em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\">u<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0574em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3714em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\">u<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 1em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0574em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3714em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\">u<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0574em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3714em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 1em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">G<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0574em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3714em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0574em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3714em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u9762\u79ef\u5143\u7d20&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">dA&#061;EG\u2212F2\u2009du\u2009dv<br \/>\ndA &#061; \\\\sqrt{EG &#8211; F^2} \\\\, du \\\\, dv<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6944em\"><\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.24em;vertical-align: -0.1777em\"><\/span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.0623em\"><span class=\"svg-align\" style=\"top: -3.2em\"><span class=\"pstrut\" style=\"height: 3.2em\"><\/span><span class=\"mord\" style=\"padding-left: 1em\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">E<\/span><span class=\"mord mathnormal\">G<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7401em\"><span class=\"\" style=\"top: -2.989em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.0223em\"><span class=\"pstrut\" style=\"height: 3.2em\"><\/span><span class=\"hide-tail\" style=\"min-width: 1.02em;height: 1.28em\"><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h4>13.5.6 \u7b2c\u4e8c\u57fa\u672c\u5f62\u5f0f&#xff08;\u66f2\u7387&#xff09;<\/h4>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">L&#061;n\u22c5\u22022S\u2202u2,M&#061;n\u22c5\u22022S\u2202u\u2202v,N&#061;n\u22c5\u22022S\u2202v2<br \/>\nL &#061; \\\\mathbf{n} \\\\cdot \\\\frac{\\\\partial^2 S}{\\\\partial u^2}, \\\\quad<br \/>\nM &#061; \\\\mathbf{n} \\\\cdot \\\\frac{\\\\partial^2 S}{\\\\partial u \\\\partial v}, \\\\quad<br \/>\nN &#061; \\\\mathbf{n} \\\\cdot \\\\frac{\\\\partial^2 S}{\\\\partial v^2}<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\">L<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.4445em\"><\/span><span class=\"mord mathbf\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.1771em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.4911em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord\"><span class=\"mord mathnormal\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7401em\"><span class=\"\" style=\"top: -2.989em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 1em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.4445em\"><\/span><span class=\"mord mathbf\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.1771em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.4911em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 1em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.4445em\"><\/span><span class=\"mord mathbf\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.1771em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.4911em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7401em\"><span class=\"\" style=\"top: -2.989em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.0556em\">\u2202<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u9ad8\u65af\u66f2\u7387\u4e0e\u5e73\u5747\u66f2\u7387&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">K&#061;LN\u2212M2EG\u2212F2,H&#061;EN\u22122FM&#043;GL2(EG\u2212F2)<br \/>\nK &#061; \\\\frac{LN &#8211; M^2}{EG &#8211; F^2}, \\\\qquad<br \/>\nH &#061; \\\\frac{EN &#8211; 2FM &#043; GL}{2(EG &#8211; F^2)}<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">K<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.2604em;vertical-align: -0.7693em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.4911em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">E<\/span><span class=\"mord mathnormal\">G<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7401em\"><span class=\"\" style=\"top: -2.989em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">L<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">M<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7693em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 2em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0813em\">H<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.2963em;vertical-align: -0.936em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3603em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">E<\/span><span class=\"mord mathnormal\">G<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7401em\"><span class=\"\" style=\"top: -2.989em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">E<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathnormal\">G<\/span><span class=\"mord mathnormal\">L<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.936em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u4e3b\u66f2\u7387&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">k1,k2&#061;H\u00b1H2\u2212K<br \/>\nk_1, k_2 &#061; H \\\\pm \\\\sqrt{H^2 &#8211; K}<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0315em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0315em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7667em;vertical-align: -0.0833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0813em\">H<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u00b1<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.24em;vertical-align: -0.1777em\"><\/span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.0623em\"><span class=\"svg-align\" style=\"top: -3.2em\"><span class=\"pstrut\" style=\"height: 3.2em\"><\/span><span class=\"mord\" style=\"padding-left: 1em\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0813em\">H<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7401em\"><span class=\"\" style=\"top: -2.989em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">K<\/span><\/span><\/span><span class=\"\" style=\"top: -3.0223em\"><span class=\"pstrut\" style=\"height: 3.2em\"><\/span><span class=\"hide-tail\" style=\"min-width: 1.02em;height: 1.28em\"><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h4>13.5.7 \u9762\u9762\u79ef<\/h4>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">A&#061;\u222cDEG\u2212F2\u2009du\u2009dv<br \/>\nA &#061; \\\\iint_D \\\\sqrt{EG &#8211; F^2} \\\\, du \\\\, dv<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.2717em;vertical-align: -0.9117em\"><\/span><span class=\"mop\"><span class=\"mop op-symbol large-op\" style=\"margin-right: 0.4445em;position: relative;top: -0.001em\">\u222c<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"\" style=\"top: -1.7883em;margin-left: -0.4445em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0278em\">D<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9117em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.0623em\"><span class=\"svg-align\" style=\"top: -3.2em\"><span class=\"pstrut\" style=\"height: 3.2em\"><\/span><span class=\"mord\" style=\"padding-left: 1em\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">E<\/span><span class=\"mord mathnormal\">G<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7401em\"><span class=\"\" style=\"top: -2.989em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.0223em\"><span class=\"pstrut\" style=\"height: 3.2em\"><\/span><span class=\"hide-tail\" style=\"min-width: 1.02em;height: 1.28em\"><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u6570\u503c\u79ef\u5206&#xff08;Gauss&#xff09;\u5728\u53c2\u6570\u57df <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">DD<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><\/span><\/span><\/span><\/span> \u4e0a\u3002<\/p>\n<h4>13.5.8 \u70b9\u5728\u9762\u5185<\/h4>\n<p><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">p\\\\mathbf{p}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6389em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathbf\">p<\/span><\/span><\/span><\/span><\/span> \u5728\u9762 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">FF<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><\/span><\/span><\/span><\/span> \u5185 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\u2005\u200a\u27fa\u2005\u200a\\\\iff<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.549em;vertical-align: -0.024em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u27fa<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><\/span><\/span><\/span> <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">S\u22121(p)\u2208DS^{-1}(\\\\mathbf{p}) \\\\in D<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathbf\">p<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2208<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><\/span><\/span><\/span><\/span>\u3002\u5373\u53c2\u6570 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">(u,v)(u,v)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u5728\u5916\u73af\u5185\u4e14\u6240\u6709\u5185\u73af\u5916\u3002\u7528\u7ed5\u6570\u5224\u5b9a\u3002<\/p>\n<h4>13.5.9 \u9762\u5230\u70b9\u8ddd\u79bb<\/h4>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">d&#061;min\u2061(u,v)\u2208D\u2223S(u,v)\u2212p\u2223<br \/>\nd &#061; \\\\min_{(u,v) \\\\in D} \\\\left| S(u,v) &#8211; \\\\mathbf{p} \\\\right|<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6944em\"><\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.716em;vertical-align: -0.966em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6679em\"><span class=\"\" style=\"top: -2.309em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">u<\/span><span class=\"mpunct mtight\">,<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose mtight\">)<\/span><span class=\"mrel mtight\">\u2208<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0278em\">D<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"\"><span class=\"mop\">min<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.966em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathbf\">p<\/span><span class=\"mclose delimcenter\" style=\"top: 0em\">\u2223<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u725b\u987f\u8fed\u4ee3\u4ece\u91c7\u6837\u521d\u503c\u4f18\u5316\u3002<\/p>\n<h4>13.5.10 p-curve<\/h4>\n<p>\u8fb9 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">ee<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">e<\/span><\/span><\/span><\/span><\/span> \u66f2\u7ebf <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">C(t)C(t)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">t<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u9762 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">FF<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><\/span><\/span><\/span><\/span> \u66f2\u9762 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">S(u,v)S(u,v)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002p-curve&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">(u(t),v(t))&#061;S\u22121(C(t))<br \/>\n(u(t), v(t)) &#061; S^{-1}(C(t))<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">t<\/span><span class=\"mclose\">)<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">t<\/span><span class=\"mclose\">))<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.1141em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">t<\/span><span class=\"mclose\">))<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5373 3D \u66f2\u7ebf\u5728\u53c2\u6570\u57df\u7684\u50cf\u3002<\/p>\n<h4>13.5.11 \u6b27\u62c9\u516c\u5f0f\u4e2d\u7684\u9762<\/h4>\n<p><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">FF<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><\/span><\/span><\/span><\/span> \u662f\u9762\u6570&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">V\u2212E&#043;F\u2212L&#061;2(S\u2212G)<br \/>\nV &#8211; E &#043; F &#8211; L &#061; 2(S &#8211; G)<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7667em;vertical-align: -0.0833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.2222em\">V<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7667em;vertical-align: -0.0833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">E<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7667em;vertical-align: -0.0833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\">L<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">2<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">G<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u52a0\u9762&#xff08;mef&#xff09;\u4f7f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">F&#043;1F&#043;1<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7667em;vertical-align: -0.0833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<hr \/>\n<h3>13.6 \u7b97\u6cd5\u4e0e\u6570\u636e\u7ed3\u6784\u8be6\u89e3<\/h3>\n<h4>13.6.1 \u9762\u6570\u636e\u7ed3\u6784<\/h4>\n<p>class Face(TShape):<br \/>\n    surface: Geom_Surface       # \u66f2\u9762\u51e0\u4f55\u652f\u6491<br \/>\n    wires: List[TopoDS_Wire]    # \u5916\u73af &#043; \u5185\u73af<br \/>\n    tolerance: float<\/p>\n<p>class TopoDS_Face(Shape):<br \/>\n    tshape: Handle[Face]<br \/>\n    location: Location<br \/>\n    orientation: Orientation    # FORWARD\/REVERSED<\/p>\n<h4>13.6.2 \u521b\u5efa\u9762<\/h4>\n<p>function makeFace(surface, outerWire, innerWires&#061;[]):<br \/>\n    tf &#061; Face(surface, [outerWire] &#043; innerWires)<br \/>\n    return TopoDS_Face(tshape&#061;tf, orientation&#061;FORWARD)<\/p>\n<h4>13.6.3 \u9762\u4e0a\u6c42\u503c\u4e0e\u6cd5\u5411<\/h4>\n<p>function faceValue(face, u, v):<br \/>\n    surface &#061; BRep_Tool.Surface(face)<br \/>\n    return surface.value(u, v)<\/p>\n<p>function faceNormal(face, u, v):<br \/>\n    surface &#061; BRep_Tool.Surface(face)<br \/>\n    Su, Sv &#061; surface.d1(u, v)  # \u504f\u5bfc<br \/>\n    n &#061; Su.cross(Sv).normalize()<br \/>\n    if face.orientation &#061;&#061; REVERSED:<br \/>\n        n &#061; -n<br \/>\n    return n<\/p>\n<h4>13.6.4 \u70b9\u5728\u9762\u5185<\/h4>\n<p>function pointInFace(p, face):<br \/>\n    surface &#061; BRep_Tool.Surface(face)<br \/>\n    (u, v) &#061; surface.inverse(p)  # \u6c42\u9006<br \/>\n    outerWire &#061; face.outerWire<br \/>\n    innerWires &#061; face.wires[1:]<br \/>\n    if not pointInWireParam((u,v), outerWire, surface):<br \/>\n        return False  # \u4e0d\u5728\u5916\u73af\u5185<br \/>\n    for inner in innerWires:<br \/>\n        if pointInWireParam((u,v), inner, surface):<br \/>\n            return False  # \u5728\u5185\u73af\u5185&#xff08;\u5b54&#xff09;<br \/>\n    return True<\/p>\n<h4>13.6.5 \u9762\u9762\u79ef<\/h4>\n<p>function faceArea(face):<br \/>\n    surface &#061; BRep_Tool.Surface(face)<br \/>\n    # \u5728\u53c2\u6570\u57df D \u4e0a\u6570\u503c\u79ef\u5206 \u221a(EG-F\u00b2)<br \/>\n    D &#061; paramDomain(face)  # \u5916\u73af &#8211; \u5185\u73af<br \/>\n    def integrand(u, v):<br \/>\n        Su, Sv &#061; surface.d1(u, v)<br \/>\n        E &#061; Su.dot(Su); F &#061; Su.dot(Sv); G &#061; Sv.dot(Sv)<br \/>\n        return sqrt(E*G &#8211; F*F)<br \/>\n    return gaussIntegrate(integrand, D)<\/p>\n<h4>13.6.6 \u9762\u5230\u70b9\u8ddd\u79bb<\/h4>\n<p>function faceToPointDistance(face, p):<br \/>\n    surface &#061; BRep_Tool.Surface(face)<br \/>\n    # \u91c7\u6837\u53c2\u6570\u57df\u627e\u6700\u8fd1\u521d\u503c<br \/>\n    (u0, v0) &#061; sampleClosest(surface, p, face)<br \/>\n    # \u725b\u987f\u8fed\u4ee3\u4f18\u5316 min |S(u,v) &#8211; p|<br \/>\n    (u, v) &#061; newtonMinimize(lambda u,v: (surface.value(u,v) &#8211; p).lengthSquared(), u0, v0)<br \/>\n    return (surface.value(u, v) &#8211; p).length()<\/p>\n<h4>13.6.7 \u5e03\u5c14\u5207\u5206\u9762<\/h4>\n<p>function splitFace(face, interCurves):<br \/>\n    # interCurves \u662f\u6c42\u4ea4\u4ea7\u751f\u7684\u4ea4\u7ebf<br \/>\n    newEdges &#061; [makeEdge(c) for c in interCurves]<br \/>\n    # \u7528 mef \u52a0\u8fb9\u628a\u9762\u5207\u5206\u4e3a\u7247\u6bb5<br \/>\n    fragments &#061; [face]<br \/>\n    for e in newEdges:<br \/>\n        newFragments &#061; []<br \/>\n        for f in fragments:<br \/>\n            split &#061; mef(f, e)  # \u52a0\u8fb9\u5206\u88c2\u9762<br \/>\n            newFragments.extend(split)<br \/>\n        fragments &#061; newFragments<br \/>\n    return fragments<\/p>\n<hr \/>\n<h3>13.7 \u4e3b\u6d41\u8f6f\u4ef6\u5b9e\u73b0\u5bf9\u6bd4<\/h3>\n<table>\n<tr>\u5185\u6838\u9762\u8868\u793a\u67e5\u8be2 API\u66f2\u9762\u7c7b\u578b\u4fee\u590d<\/tr>\n<tbody>\n<tr>\n<td>OCCT<\/td>\n<td>TopoDS_Face &#043; BRep_TFace &#043; Geom_Surface<\/td>\n<td>BRep_Tool::Surface\/Tolerance&#xff1b;BRepAdaptor_Surface \u6c42\u503c\u6c42\u5bfc<\/td>\n<td>Geom_Plane\/CylindricalSurface\/BSplineSurface \u7b49<\/td>\n<td>ShapeFix_Face<\/td>\n<\/tr>\n<tr>\n<td>Parasolid<\/td>\n<td>FACE \u5b9e\u4f53\u5e26\u66f2\u9762\u4e0e\u73af&#xff0c;\u72b6\u6001\u673a\u7ba1\u7406<\/td>\n<td>PK_FACE_ask \u67e5\u8be2<\/td>\n<td>\u89e3\u6790 &#043; NURBS<\/td>\n<td>\u5185\u7f6e\u4fee\u590d<\/td>\n<\/tr>\n<tr>\n<td>CGM<\/td>\n<td>\u5bb9\u5dee\u9762&#xff0c;\u5bb9\u5dee\u5efa\u6a21\u8d2f\u7a7f<\/td>\n<td>\u2014<\/td>\n<td>\u89e3\u6790 &#043; NURBS<\/td>\n<td>\u5bb9\u5dee\u9a71\u52a8<\/td>\n<\/tr>\n<tr>\n<td>ACIS<\/td>\n<td>FACE \u5e26 surface \u4e0e loop&#xff0c;\u8f90\u5c04\u8fb9\u5c42<\/td>\n<td>\u2014<\/td>\n<td>\u89e3\u6790 &#043; NURBS<\/td>\n<td>\u2014<\/td>\n<\/tr>\n<tr>\n<td>Granite<\/td>\n<td>\u9762\u5d4c\u5165\u7279\u5f81\u5386\u53f2<\/td>\n<td>\u2014<\/td>\n<td>\u89e3\u6790 &#043; NURBS<\/td>\n<td>\u2014<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u5404\u5b9e\u73b0\u90fd\u628a\u9762\u4f5c\u66f2\u9762 &#043; \u73af &#043; \u5bb9\u5dee &#043; \u65b9\u5411&#xff0c;\u5dee\u5f02\u5728\u66f2\u9762\u7c7b\u578b\u4e0e\u5bb9\u5dee\u3002<\/p>\n<hr \/>\n<h3>13.8 \u64cd\u4f5c\u6b65\u9aa4\u6559\u7a0b<\/h3>\n<p>\u7528 FreeCAD&#xff08;OCCT&#xff09;\u4f53\u9a8c\u9762\u3002<\/p>\n<h4>\u6b65\u9aa4\u4e00&#xff1a;\u67e5\u770b\u9762<\/h4>\n<p><span class=\"token keyword\">import<\/span> Part<br \/>\nbox <span class=\"token operator\">&#061;<\/span> Part<span class=\"token punctuation\">.<\/span>makeBox<span class=\"token punctuation\">(<\/span><span class=\"token number\">10<\/span><span class=\"token punctuation\">,<\/span><span class=\"token number\">10<\/span><span class=\"token punctuation\">,<\/span><span class=\"token number\">10<\/span><span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token keyword\">for<\/span> f <span class=\"token keyword\">in<\/span> box<span class=\"token punctuation\">.<\/span>Faces<span class=\"token punctuation\">:<\/span><br \/>\n    <span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span>f<span class=\"token punctuation\">.<\/span>Area<span class=\"token punctuation\">,<\/span> f<span class=\"token punctuation\">.<\/span>CenterOfMass<span class=\"token punctuation\">,<\/span> f<span class=\"token punctuation\">.<\/span>normalAt<span class=\"token punctuation\">(<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;Surface:&#034;<\/span><span class=\"token punctuation\">,<\/span> f<span class=\"token punctuation\">.<\/span>Surface<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;Wires:&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token builtin\">len<\/span><span class=\"token punctuation\">(<\/span>f<span class=\"token punctuation\">.<\/span>Wires<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><\/p>\n<h4>\u6b65\u9aa4\u4e8c&#xff1a;\u67e5\u8be2\u66f2\u9762\u7c7b\u578b<\/h4>\n<p>f <span class=\"token operator\">&#061;<\/span> box<span class=\"token punctuation\">.<\/span>Faces<span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span><br \/>\n<span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span><span class=\"token builtin\">type<\/span><span class=\"token punctuation\">(<\/span>f<span class=\"token punctuation\">.<\/span>Surface<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span>  <span class=\"token comment\"># Geom_Plane<\/span><br \/>\n<span class=\"token comment\"># \u5706\u67f1\u9762<\/span><br \/>\ncyl <span class=\"token operator\">&#061;<\/span> Part<span class=\"token punctuation\">.<\/span>makeCylinder<span class=\"token punctuation\">(<\/span><span class=\"token number\">5<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">10<\/span><span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span><span class=\"token builtin\">type<\/span><span class=\"token punctuation\">(<\/span>cyl<span class=\"token punctuation\">.<\/span>Faces<span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">.<\/span>Surface<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span>  <span class=\"token comment\"># Geom_CylindricalSurface<\/span><\/p>\n<h4>\u6b65\u9aa4\u4e09&#xff1a;\u9762\u4e0a\u6c42\u503c\u4e0e\u6cd5\u5411<\/h4>\n<p>f <span class=\"token operator\">&#061;<\/span> box<span class=\"token punctuation\">.<\/span>Faces<span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span><br \/>\n<span class=\"token comment\"># \u5728\u53c2\u6570 (u,v) \u6c42\u503c<\/span><br \/>\np <span class=\"token operator\">&#061;<\/span> f<span class=\"token punctuation\">.<\/span>valueAt<span class=\"token punctuation\">(<\/span><span class=\"token number\">0.5<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0.5<\/span><span class=\"token punctuation\">)<\/span>  <span class=\"token comment\"># \u53c2\u6570\u57df\u4e2d\u70b9<\/span><br \/>\nn <span class=\"token operator\">&#061;<\/span> f<span class=\"token punctuation\">.<\/span>normalAt<span class=\"token punctuation\">(<\/span><span class=\"token number\">0.5<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0.5<\/span><span class=\"token punctuation\">)<\/span>  <span class=\"token comment\"># \u6cd5\u5411<\/span><br \/>\n<span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span>p<span class=\"token punctuation\">,<\/span> n<span class=\"token punctuation\">)<\/span><\/p>\n<h4>\u6b65\u9aa4\u56db&#xff1a;\u70b9\u5728\u9762\u5185<\/h4>\n<p>f <span class=\"token operator\">&#061;<\/span> box<span class=\"token punctuation\">.<\/span>Faces<span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span><br \/>\np <span class=\"token operator\">&#061;<\/span> Base<span class=\"token punctuation\">.<\/span>Vector<span class=\"token punctuation\">(<\/span><span class=\"token number\">5<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">5<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">)<\/span>  <span class=\"token comment\"># \u9762\u4e0a\u4e00\u70b9<\/span><br \/>\n<span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span>f<span class=\"token punctuation\">.<\/span>isInside<span class=\"token punctuation\">(<\/span>p<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span>  <span class=\"token comment\"># True<\/span><\/p>\n<h4>\u6b65\u9aa4\u4e94&#xff1a;\u5e26\u5b54\u9762\u67e5\u770b\u5185\u73af<\/h4>\n<p><span class=\"token comment\"># \u5728\u9762\u4e0a\u6253\u5b54\u540e<\/span><br \/>\nf <span class=\"token operator\">&#061;<\/span> holed<span class=\"token punctuation\">.<\/span>Faces<span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span><br \/>\n<span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span><span class=\"token builtin\">len<\/span><span class=\"token punctuation\">(<\/span>f<span class=\"token punctuation\">.<\/span>Wires<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span>  <span class=\"token comment\"># \u5916\u73af &#043; \u5185\u73af<\/span><\/p>\n<h4>\u6b65\u9aa4\u516d&#xff1a;\u9762\u9762\u79ef\u4e0e\u66f2\u7387<\/h4>\n<p>f <span class=\"token operator\">&#061;<\/span> cyl<span class=\"token punctuation\">.<\/span>Faces<span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span>  <span class=\"token comment\"># \u5706\u67f1\u9762<\/span><br \/>\n<span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span>f<span class=\"token punctuation\">.<\/span>Area<span class=\"token punctuation\">)<\/span>  <span class=\"token comment\"># 2\u03c0rh<\/span><br \/>\n<span class=\"token comment\"># \u66f2\u7387&#xff08;\u5706\u67f1\u9762\u9ad8\u65af\u66f2\u7387 0&#xff0c;\u4e3b\u66f2\u7387 1\/r, 0&#xff09;<\/span><\/p>\n<h4>\u6b65\u9aa4\u4e03&#xff1a;\u4f53\u9a8c\u4fee\u526a\u66f2\u9762<\/h4>\n<p>\u521b\u5efa\u4e00\u4e2a NURBS \u66f2\u9762&#xff0c;\u7528\u73af\u4fee\u526a&#xff0c;\u67e5\u770b\u53c2\u6570\u57df <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">DD<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<hr \/>\n<h3>13.9 \u5e94\u7528\u573a\u666f\u4e0e\u6848\u4f8b<\/h3>\n<table>\n<tr>\u573a\u666f\u9762\u7684\u89d2\u8272<\/tr>\n<tbody>\n<tr>\n<td>\u4f53\u7d20\u521b\u5efa<\/td>\n<td>\u7acb\u65b9\u4f53 6 \u5e73\u9762\u9762\u3001\u5706\u67f1 1 \u5706\u67f1\u9762 &#043; 2 \u5e73\u9762\u9762\u7b49<\/td>\n<\/tr>\n<tr>\n<td>\u5e03\u5c14\u8fd0\u7b97<\/td>\n<td>\u6c42\u4ea4\u4ea7\u751f\u65b0\u8fb9&#xff0c;\u65b0\u8fb9\u5207\u5206\u9762\u4e3a\u7247\u6bb5&#xff0c;\u6309\u8fd0\u7b97\u7c7b\u578b\u9009\u7247\u6bb5\u7ec4\u6210\u65b0\u9762<\/td>\n<\/tr>\n<tr>\n<td>\u6e32\u67d3<\/td>\n<td>\u9762\u79bb\u6563\u6210\u4e09\u89d2\u7f51\u683c\u9001 GPU&#xff1b;\u9762\u6cd5\u5411\u7528\u4e8e\u7740\u8272<\/td>\n<\/tr>\n<tr>\n<td>\u7f51\u683c\u5316<\/td>\n<td>FEA \u7f51\u683c\u5668\u79bb\u6563\u9762\u6210\u4e09\u89d2\u7f51\u683c&#xff0c;\u6cbf\u73af\u8fb9\u4fdd\u8bc1\u8282\u70b9\u5339\u914d<\/td>\n<\/tr>\n<tr>\n<td>\u8d28\u91cf\u8ba1\u7b97<\/td>\n<td>\u9762\u9762\u79ef\u7528\u79ef\u5206&#xff0c;\u4f53\u4f53\u79ef\u7528\u6563\u5ea6\u5b9a\u7406\u9010\u9762\u79ef\u5206<\/td>\n<\/tr>\n<tr>\n<td>\u5de5\u7a0b\u56fe<\/td>\n<td>\u9762\u6295\u5f71\u5230\u5e73\u9762&#xff0c;HLR \u6d88\u9664\u9690\u85cf\u9762\u8fb9<\/td>\n<\/tr>\n<tr>\n<td>\u7279\u5f81\u8bc6\u522b<\/td>\n<td>\u5e73\u9762\u9762\u3001\u5706\u67f1\u9762\u3001\u81ea\u7531\u66f2\u9762\u9762\u8bc6\u522b\u652f\u6301\u7279\u5f81\u8bc6\u522b\u4e0e\u76f4\u63a5\u5efa\u6a21<\/td>\n<\/tr>\n<tr>\n<td>\u76f4\u63a5\u5efa\u6a21<\/td>\n<td>\u79fb\u52a8\u9762\u3001\u66ff\u6362\u9762\u3001\u5220\u9664\u9762\u7b49\u76f4\u63a5\u7f16\u8f91\u64cd\u4f5c\u9762<\/td>\n<\/tr>\n<tr>\n<td>\u6570\u636e\u4ea4\u6362<\/td>\n<td>STEP \u628a\u9762\u5e8f\u5217\u5316\u4e3a ADVANCED_FACE&#xff08;\u66f2\u9762 &#043; \u73af&#xff09;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>13.9.1 \u6563\u5ea6\u5b9a\u7406\u7684\u4f53\u79ef\u8ba1\u7b97<\/h4>\n<p>\u8bbe\u4f53 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">VV<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.2222em\">V<\/span><\/span><\/span><\/span><\/span> \u8fb9\u754c\u4e3a\u9762\u96c6 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">{Fi}\\\\{F_i\\\\}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">{<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.1389em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5219&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">Vol(V)&#061;13\u2211i\u222cFip\u22c5n\u2009dA<br \/>\n\\\\text{Vol}(V) &#061; \\\\frac{1}{3} \\\\sum_i \\\\iint_{F_i} \\\\mathbf{p} \\\\cdot \\\\mathbf{n} \\\\, dA<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord text\"><span class=\"mord\">Vol<\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.2222em\">V<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.6377em;vertical-align: -1.2777em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">3<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.05em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\"><span class=\"mop op-symbol large-op\" style=\"margin-right: 0.4445em;position: relative;top: -0.001em\">\u222c<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"\" style=\"top: -1.7883em;margin-left: -0.4445em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.1389em\">F<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3281em\"><span class=\"\" style=\"top: -2.357em;margin-left: -0.1389em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.143em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.0118em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathbf\">p<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6944em\"><\/span><span class=\"mord mathbf\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">A<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5176\u4e2d <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">p\\\\mathbf{p}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6389em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathbf\">p<\/span><\/span><\/span><\/span><\/span> \u4e3a\u9762\u4e0a\u70b9\u4f4d\u7f6e&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">n\\\\mathbf{n}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4444em\"><\/span><span class=\"mord mathbf\">n<\/span><\/span><\/span><\/span><\/span> \u4e3a\u9762\u6cd5\u5411\u3002<\/p>\n<hr \/>\n<h3>13.10 \u5e38\u89c1\u95ee\u9898\u4e0e\u8c03\u4f18<\/h3>\n<table>\n<tr>\u95ee\u9898\u539f\u56e0\u8c03\u4f18<\/tr>\n<tbody>\n<tr>\n<td>\u9762\u81ea\u4ea4<\/td>\n<td>\u66f2\u9762\u81ea\u4ea4\u5bfc\u81f4\u5e03\u5c14\u3001\u7f51\u683c\u5316\u5f02\u5e38<\/td>\n<td>ShapeFix_Face \u68c0\u6d4b\u4fee\u590d&#xff1b;\u91cd\u5efa\u6a21<\/td>\n<\/tr>\n<tr>\n<td>\u9762\u6cd5\u5411\u4e0d\u4e00\u81f4<\/td>\n<td>\u4e2a\u522b\u9762 orientation \u9519\u5bfc\u81f4\u6e32\u67d3\u6216\u5e03\u5c14\u5f02\u5e38<\/td>\n<td>ShapeFix_Face \u4fee\u590d\u6cd5\u5411&#xff1b;\u624b\u52a8\u7ffb\u8f6c<\/td>\n<\/tr>\n<tr>\n<td>\u9762\u672a\u95ed\u5408<\/td>\n<td>\u73af\u4e0d\u95ed\u5408\u5bfc\u81f4\u9762\u8fb9\u754c\u6709\u7f3a\u53e3<\/td>\n<td>ShapeFix_Wire \u4fee\u590d\u73af\u95ed\u5408<\/td>\n<\/tr>\n<tr>\n<td>\u53c2\u6570\u57df\u9519\u8bef<\/td>\n<td>\u5916\u73af <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\u2212-<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6667em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">\u2212<\/span><\/span><\/span><\/span><\/span> \u5185\u73af\u5b9a\u4e49\u7684 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">DD<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><\/span><\/span><\/span><\/span> \u4e0d\u5bf9<\/td>\n<td>ShapeFix_Face \u4fee\u590d\u53c2\u6570\u57df<\/td>\n<\/tr>\n<tr>\n<td>p-curve \u4e0d\u4e00\u81f4<\/td>\n<td>p-curve \u4e0e 3D \u66f2\u7ebf\u4e0d\u5339\u914d<\/td>\n<td>ShapeFix_Edge \u91cd\u7b97 p-curve<\/td>\n<\/tr>\n<tr>\n<td>\u66f2\u9762\u9000\u5316<\/td>\n<td>\u66f2\u9762\u5728\u67d0\u5904\u9000\u5316&#xff08;\u5982 NURBS \u8fb9\u754c&#xff09;<\/td>\n<td>ShapeUpgrade \u4fee\u590d\u9000\u5316<\/td>\n<\/tr>\n<tr>\n<td>\u9762\u5230\u8fb9\u8ddd\u79bb\u8d85\u5bb9\u5dee<\/td>\n<td>\u8fb9\u672a\u8d34\u5408\u9762<\/td>\n<td>\u8c03\u5bb9\u5dee&#xff1b;ShapeFix \u4fee\u590d<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<hr \/>\n<h3>13.11 \u53d1\u5c55\u8d8b\u52bf\u4e0e\u5c55\u671b<\/h3>\n<li>T \u6837\u6761\u9762&#xff1a;T-splines \u652f\u6301\u5c40\u90e8\u7ec6\u5316&#xff0c;\u53ef\u80fd\u5e76\u5165\u9762\u51e0\u4f55\u652f\u6491&#xff1b;<\/li>\n<li>\u5bb9\u5dee\u9762\u6df1\u5316&#xff1a;\u5bb9\u5dee\u5efa\u6a21\u666e\u53ca&#xff0c;\u9762\u5bb9\u5dee\u66f4\u81ea\u9002\u5e94&#xff1b;<\/li>\n<li>\u9762\u4fee\u590d\u81ea\u52a8\u5316&#xff1a;ShapeFix \u81ea\u52a8\u68c0\u6d4b\u4e0e\u4fee\u590d\u9762\u81ea\u4ea4\u3001\u9000\u5316\u3001\u9519\u5411&#xff1b;<\/li>\n<li>\u9762\u5728\u76f4\u63a5\u5efa\u6a21&#xff1a;\u76f4\u63a5\u7f16\u8f91\u9762&#xff08;\u79fb\u52a8\u3001\u66ff\u6362\u3001\u5220\u9664&#xff09;\u66f4\u667a\u80fd&#xff1b;<\/li>\n<li>\u5e76\u884c\u9762\u5904\u7406&#xff1a;\u9762\u72ec\u7acb\u53ef\u5e76\u884c\u6c42\u503c\u3001\u79ef\u5206\u3001\u7f51\u683c\u5316&#xff0c;GPU \u52a0\u901f&#xff1b;<\/li>\n<li>\u53ef\u5fae\u9762&#xff1a;\u53ef\u5fae\u66f2\u9762\u652f\u6301\u53ef\u5fae\u51e0\u4f55\u4e0e\u751f\u6210\u5f0f\u8bbe\u8ba1\u3002<\/li>\n<hr \/>\n<h3>13.12 \u5c0f\u7ed3<\/h3>\n<p>\u9762\u662f B-Rep \u62d3\u6251\u5c42\u6b21\u7684\u4e8c\u7ef4\u5b9e\u4f53&#xff0c;\u5173\u8054\u66f2\u9762&#xff08;\u51e0\u4f55\u652f\u6491&#xff09;&#043; \u5916\u73af &#043; \u5185\u73af &#043; \u65b9\u5411 &#043; \u5bb9\u5dee\u3002\u9762\u53c2\u6570\u57df <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">DD<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><\/span><\/span><\/span><\/span> \u7531\u5916\u73af <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\u2212-<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6667em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">\u2212<\/span><\/span><\/span><\/span><\/span> \u5185\u73af\u5b9a\u4e49&#xff0c;\u9762\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">S(D)S(D)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002\u9762\u6cd5\u5411\u7531 orientation \u4e0e\u66f2\u9762\u81ea\u7136\u6cd5\u5411\u51b3\u5b9a\u3002\u9762\u652f\u6301\u6c42\u503c\u3001\u6c42\u5bfc\u3001\u66f2\u7387\u3001\u9762\u79ef\u3001\u70b9\u5206\u7c7b\u3002\u9762\u5728\u5e03\u5c14\u4e2d\u88ab\u5207\u5206&#xff0c;\u5728\u6e32\u67d3\u3001\u7f51\u683c\u5316\u3001\u8d28\u91cf\u8ba1\u7b97\u3001\u5de5\u7a0b\u56fe\u4e2d\u6838\u5fc3\u3002\u9762\u5e26\u5bb9\u5dee\u7528\u4e8e\u8d34\u5408\u5224\u5b9a&#xff0c;p-curve \u652f\u6301\u9762\u4e0a\u64cd\u4f5c\u3002\u9762\u81ea\u4ea4\u3001\u9000\u5316\u3001\u9519\u5411\u9700 ShapeFix \u4fee\u590d\u3002\u5404\u5185\u6838\u90fd\u628a\u9762\u4f5c\u66f2\u9762 &#043; \u73af &#043; \u5bb9\u5dee&#xff0c;\u5dee\u5f02\u5728\u66f2\u9762\u7c7b\u578b\u4e0e\u5bb9\u5dee\u3002\u7406\u89e3\u9762\u662f\u7406\u89e3 B-Rep \u51e0\u4f55\u8868\u8fbe\u7684\u6838\u5fc3&#xff0c;\u4e3a\u7b2c 14 \u7ae0\u58f3\u4e0e\u4f53\u94fa\u57ab\u3002<\/p>\n<h4>13.12.1 \u5173\u952e\u516c\u5f0f\u901f\u67e5<\/h4>\n<table>\n<tr>\u91cf\u516c\u5f0f<\/tr>\n<tbody>\n<tr>\n<td>\u9762\u53c2\u6570\u5316<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">F&#061;S(D)F &#061; S(D)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">D&#061;Douter\u2216\u22c3DinnerD &#061; D_{\\\\text{outer}} \\\\setminus \\\\bigcup D_{\\\\text{inner}}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2806em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord text mtight\"><span class=\"mord mtight\">outer<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2216<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u22c3<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3175em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord text mtight\"><span class=\"mord mtight\">inner<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>\u9762\u6cd5\u5411<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">n&#061;orientation\u00d7Su\u00d7Sv\u2223Su\u00d7Sv\u2223\\\\mathbf{n} &#061; \\\\text{orientation} \\\\times \\\\dfrac{S_u \\\\times S_v}{|S_u \\\\times S_v|}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4444em\"><\/span><span class=\"mord mathbf\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7512em;vertical-align: -0.0833em\"><\/span><span class=\"mord text\"><span class=\"mord\">orientation<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.2963em;vertical-align: -0.936em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3603em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">\u2223<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0576em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">u<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0576em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0359em\">v<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\u2223<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0576em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">u<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0576em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0359em\">v<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.936em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>\u7b2c\u4e00\u57fa\u672c\u5f62\u5f0f<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">E&#061;Su\u22c5SuE &#061; S_u \\\\cdot S_u<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">E<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0576em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">u<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0576em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">u<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">F&#061;Su\u22c5SvF &#061; S_u \\\\cdot S_v<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0576em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">u<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0576em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0359em\">v<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">G&#061;Sv\u22c5SvG &#061; S_v \\\\cdot S_v<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\">G<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0576em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0359em\">v<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0576em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0359em\">v<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>\u9762\u79ef\u5143\u7d20<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">dA&#061;EG\u2212F2\u2009du\u2009dvdA &#061; \\\\sqrt{EG &#8211; F^2} \\\\, du \\\\, dv<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6944em\"><\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.04em;vertical-align: -0.1266em\"><\/span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9134em\"><span class=\"svg-align\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\" style=\"padding-left: 0.833em\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">E<\/span><span class=\"mord mathnormal\">G<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7401em\"><span class=\"\" style=\"top: -2.989em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -2.8734em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"hide-tail\" style=\"min-width: 0.853em;height: 1.08em\"><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1266em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>\u7b2c\u4e8c\u57fa\u672c\u5f62\u5f0f<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">L&#061;n\u22c5SuuL &#061; \\\\mathbf{n} \\\\cdot S_{uu}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\">L<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.4445em\"><\/span><span class=\"mord mathbf\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0576em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">uu<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">M&#061;n\u22c5SuvM &#061; \\\\mathbf{n} \\\\cdot S_{uv}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.4445em\"><\/span><span class=\"mord mathbf\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0576em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0359em\">uv<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">N&#061;n\u22c5SvvN &#061; \\\\mathbf{n} \\\\cdot S_{vv}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.4445em\"><\/span><span class=\"mord mathbf\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0576em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0359em\">v<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>\u9ad8\u65af\u66f2\u7387<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">K&#061;LN\u2212M2EG\u2212F2K &#061; \\\\dfrac{LN &#8211; M^2}{EG &#8211; F^2}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">K<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.2604em;vertical-align: -0.7693em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.4911em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">E<\/span><span class=\"mord mathnormal\">G<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7401em\"><span class=\"\" style=\"top: -2.989em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">L<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">M<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7693em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>\u5e73\u5747\u66f2\u7387<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">H&#061;EN\u22122FM&#043;GL2(EG\u2212F2)H &#061; \\\\dfrac{EN &#8211; 2FM &#043; GL}{2(EG &#8211; F^2)}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0813em\">H<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.2963em;vertical-align: -0.936em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3603em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">E<\/span><span class=\"mord mathnormal\">G<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7401em\"><span class=\"\" style=\"top: -2.989em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">E<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathnormal\">G<\/span><span class=\"mord mathnormal\">L<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.936em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>\u4e3b\u66f2\u7387<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">k1,2&#061;H\u00b1H2\u2212Kk_{1,2} &#061; H \\\\pm \\\\sqrt{H^2 &#8211; K}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.9805em;vertical-align: -0.2861em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0315em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1<\/span><span class=\"mpunct mtight\">,<\/span><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7667em;vertical-align: -0.0833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0813em\">H<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u00b1<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.04em;vertical-align: -0.1266em\"><\/span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9134em\"><span class=\"svg-align\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\" style=\"padding-left: 0.833em\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0813em\">H<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7401em\"><span class=\"\" style=\"top: -2.989em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">K<\/span><\/span><\/span><span class=\"\" style=\"top: -2.8734em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"hide-tail\" style=\"min-width: 0.853em;height: 1.08em\"><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1266em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>\u9762\u79ef<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">A&#061;\u222cDEG\u2212F2\u2009du\u2009dvA &#061; \\\\iint_D \\\\sqrt{EG &#8211; F^2} \\\\, du \\\\, dv<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.2691em;vertical-align: -0.3557em\"><\/span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"margin-right: 0.1945em;position: relative;top: -0.0005em\">\u222c<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1226em\"><span class=\"\" style=\"top: -2.3443em;margin-left: -0.1945em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0278em\">D<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3557em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9134em\"><span class=\"svg-align\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\" style=\"padding-left: 0.833em\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">E<\/span><span class=\"mord mathnormal\">G<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7401em\"><span class=\"\" style=\"top: -2.989em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -2.8734em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"hide-tail\" style=\"min-width: 0.853em;height: 1.08em\"><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1266em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>\u9762\u5230\u70b9\u8ddd\u79bb<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">d&#061;min\u2061(u,v)\u2208D\u2223S(u,v)\u2212p\u2223d &#061; \\\\min_{(u,v) \\\\in D} |S(u,v) &#8211; \\\\mathbf{p}|<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6944em\"><\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.1052em;vertical-align: -0.3552em\"><\/span><span class=\"mop\"><span class=\"mop\">min<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3448em\"><span class=\"\" style=\"top: -2.5198em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">u<\/span><span class=\"mpunct mtight\">,<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose mtight\">)<\/span><span class=\"mrel mtight\">\u2208<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0278em\">D<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3552em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathbf\">p<\/span><span class=\"mord\">\u2223<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>p-curve<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">(u(t),v(t))&#061;S\u22121(C(t))(u(t), v(t)) &#061; S^{-1}(C(t))<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">u<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">t<\/span><span class=\"mclose\">)<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">t<\/span><span class=\"mclose\">))<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">t<\/span><span class=\"mclose\">))<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>\u6b27\u62c9\u516c\u5f0f<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">V\u2212E&#043;F\u2212L&#061;2(S\u2212G)V &#8211; E &#043; F &#8211; L &#061; 2(S &#8211; G)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7667em;vertical-align: -0.0833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.2222em\">V<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7667em;vertical-align: -0.0833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">E<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7667em;vertical-align: -0.0833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\">L<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">2<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">G<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<hr \/>\n<h3>13.13 \u5ef6\u4f38\u9605\u8bfb<\/h3>\n<li>OpenCASCADE TopoDS_Face\u3001BRep_Tool\u3001BRepAdaptor_Surface\u3001Geom_Surface \u6587\u6863&#xff1b;<\/li>\n<li>Piegl L, Tiller W. The NURBS Book[M]. 2nd ed. Springer, 1997.&#xff08;\u66f2\u9762\u6c42\u503c\u6c42\u5bfc\u66f2\u7387&#xff09;<\/li>\n<li>do Carmo M P. Differential Geometry of Curves and Surfaces[M]. Prentice-Hall, 1976.&#xff08;\u7b2c\u4e00\u7b2c\u4e8c\u57fa\u672c\u5f62\u5f0f&#xff09;<\/li>\n<li>ACIS FACE \u6587\u6863&#xff1b;<\/li>\n<li>\u672c\u62a5\u544a\u7b2c 12 \u7ae0\u73af\u3001\u7b2c 14 \u7ae0\u58f3\u4e0e\u4f53\u3001\u7b2c 24\u201325 \u7ae0 NURBS \u66f2\u9762\u4e0e\u4fee\u526a\u66f2\u9762\u3002<\/li>\n<hr \/>\n<h3>\u53c2\u8003\u6587\u732e<\/h3>\n<p>[1] Open CASCADE Technology Documentation: TopoDS_Face, BRep_Tool, BRepAdaptor_Surface, Geom_Surface, OCCT 8.0, 2024.<br \/>\n[2] Piegl L, Tiller W. The NURBS Book[M]. 2nd ed. Springer, 1997.<br \/>\n[3] do Carmo M P. Differential Geometry of Curves and Surfaces[M]. Prentice-Hall, 1976.<br \/>\n[4] Requicha A A G. Representations for Rigid Solids: Theory, Methods, and Systems[J]. ACM Computing Surveys, 1980, 12(4): 437-464.<br \/>\n[5] M\u00e4ntyl\u00e4 M. An Introduction to Solid Modeling[M]. Computer Science Press, 1988.&#xff08;B-Rep \u62d3\u6251\u5c42\u6b21&#xff09;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u7b2c13\u7ae0 \u62d3\u6251\u5b9e\u4f53\u9762\u7ae0\u8282\u5b9a\u4f4d&#xff1a;B-Rep \u62d3\u6251\u5c42\u6b21\u4e2d\u8fde\u63a5\u73af\u4e0e\u58f3\u7684\u5173\u952e\u4e8c\u7ef4\u5b9e\u4f53<br \/>\n\u5173\u952e\u8bcd&#xff1a;\u62d3\u6251\u9762&#xff1b;B-Rep&#xff1b;\u53c2\u6570\u57df&#xff1b;p-curve&#xff1b;\u7b2c\u4e00\/\u7b2c\u4e8c\u57fa\u672c\u5f62\u5f0f&#xff1b;\u6cd5\u5411\u5b9a\u5411&#xff1b;\u81ea\u4ea4\u4fee\u590d\u6458\u8981<br \/>\n\u9762&#xff08;Face&#xff09;\u662f B-Rep \u62d3\u6251\u5c42\u6b21\u4e2d\u7684\u4e8c\u7ef4\u5b9e\u4f53&#xff0c;\u5173\u8054\u4e00\u4e2a\u66f2\u9762\u4f5c\u4e3a\u51e0\u4f55\u652f\u6491&#xff0c;\u7531\u5916\u73af\u4e0e\u5185\u73af\u754c\u5b9a\u5176\u5728\u53c2\u6570\u57df\u4e0a\u7684\u533a\u57df\u3002\u9762\u662f\u51e0\u4f55\u4e0e\u62d3\u6251\u4ea4\u6c47\u7684\u6838\u5fc3<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[8512,2606,292,427],"topic":[],"class_list":["post-114291","post","type-post","status-publish","format-standard","hentry","category-server","tag-cad","tag-2606","tag-292","tag-427"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.3 - 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