{"id":91920,"date":"2026-08-08T18:32:45","date_gmt":"2026-08-08T10:32:45","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/91920.html"},"modified":"2026-08-08T18:32:45","modified_gmt":"2026-08-08T10:32:45","slug":"%e6%9b%b2%e7%ba%bf%e5%a4%9f%e5%b9%b3%e5%b0%b1%e5%88%ab%e5%86%8d%e5%88%87%ef%bc%9a%e8%87%aa%e9%80%82%e5%ba%94%e8%b4%9d%e5%a1%9e%e5%b0%94%e7%a6%bb%e6%95%a3%e5%8c%96%e7%9a%84%e9%80%90%e6%ad%a5%e5%8f%af","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/91920.html","title":{"rendered":"\u66f2\u7ebf\u591f\u5e73\u5c31\u522b\u518d\u5207\uff1a\u81ea\u9002\u5e94\u8d1d\u585e\u5c14\u79bb\u6563\u5316\u7684\u9010\u6b65\u53ef\u89c6\u5316h"},"content":{"rendered":"

\u56fa\u5b9a\u6b65\u957f\u91c7\u6837\u8d1d\u585e\u5c14\u66f2\u7ebf\u8981\u4e48\u5728\u5e73\u76f4\u533a\u57df\u6d6a\u8d39\u70b9，\u8981\u4e48\u5728\u6025\u5f2f\u5904\u9732\u51fa\u6298\u7ebf\u3002\u81ea\u9002\u5e94\u79bb\u6563\u5316\u901a\u8fc7\u6bd4\u8f83\u63a7\u5236\u70b9\u5230\u7aef\u70b9\u5f26\u7ebf\u7684\u8ddd\u79bb，\u5e73\u5766\u5c31\u8f93\u51fa\u7ebf\u6bb5，\u4e0d\u5e73\u5c31\u7528 De Casteljau \u5bf9\u534a\u5207\u5206\u3002\u672c\u6587\u7528 C++17 \u5b8c\u6574\u5b9e\u73b0\u4e09\u6b21\u8d1d\u585e\u5c14\u9012\u5f52\u7ec6\u5206，\u9010\u5c42\u89e3\u91ca\u8bef\u5dee\u3001\u7ec8\u6b62\u6761\u4ef6\u4e0e\u9000\u5316\u66f2\u7ebf\u3002<\/p>\n

\u5728\u5c4f\u5e55\u4e0a\u753b\u4e09\u6b21\u8d1d\u585e\u5c14\u66f2\u7ebf，\u6700\u7ec8\u4ecd\u8981\u4ea4\u7ed9\u5149\u6805\u5668\u4e00\u4e32\u7ebf\u6bb5\u3002\u56fa\u5b9a\u53d6 100 \u4e2a t \u503c\u5f53\u7136\u80fd\u753b，\u4f46\u7f29\u653e\u540e\u53ef\u80fd\u4e0d\u591f\u5e73\u6ed1，\u7f29\u5c0f\u65f6\u53c8\u767d\u767d\u751f\u6210\u5927\u91cf\u51e0\u4e4e\u5171\u7ebf\u7684\u70b9\u3002\u66f4\u5408\u9002\u7684\u95ee\u9898\u662f：\u5f53\u524d\u8fd9\u6bb5\u66f2\u7ebf\u770b\u8d77\u6765\u662f\u5426\u5df2\u7ecf\u8db3\u591f\u50cf\u5b83\u7684\u7aef\u70b9\u8fde\u7ebf？\u82e5\u7b54\u6848\u662f，\u5c31\u505c\u6b62；\u5426\u5219\u628a\u5b83\u5207\u6210\u5de6\u53f3\u4e24\u534a\u7ee7\u7eed\u95ee\u3002<\/p>\n

\u753b\u9762\u7b2c\u4e00\u5c42：\u7aef\u70b9\u5f26\u7ebf<\/h3>\n

\u4e09\u6b21\u8d1d\u585e\u5c14\u7531 P0,P1,P2,P3 \u5b9a\u4e49，\u66f2\u7ebf\u4ece P0 \u5230 P3\u3002\u82e5\u4e24\u4e2a\u5185\u90e8\u63a7\u5236\u70b9\u90fd\u7d27\u8d34\u5f26\u7ebf P0P3，\u6574\u6bb5\u66f2\u7ebf\u63a5\u8fd1\u76f4\u7ebf\u3002\u6211\u4eec\u7528\u63a7\u5236\u70b9\u5230\u65e0\u9650\u76f4\u7ebf\u7684\u5782\u76f4\u8ddd\u79bb\u4f5c\u4e3a\u5e73\u5766\u5ea6\u6307\u6807，\u5e76\u53d6\u8f83\u5927\u8005\u3002\u8fd9\u4e2a\u6307\u6807\u7b80\u5355\u4e14\u5e38\u7528，\u4f46\u4e0d\u662f\u4e25\u683c\u7684\u50cf\u7d20\u8bef\u5dee\u4e0a\u754c；\u5bb9\u5dee\u5e94\u7ed3\u5408\u5750\u6807\u5c3a\u5ea6\u548c\u6700\u7ec8\u53d8\u6362\u9009\u62e9\u3002<\/p>\n

\u753b\u9762\u7b2c\u4e8c\u5c42：De Casteljau \u5bf9\u534a\u5207<\/h3>\n

\u4ee4\u53c2\u6570 t=0.5，\u5148\u5bf9\u76f8\u90bb\u63a7\u5236\u70b9\u53d6\u4e2d\u70b9，\u518d\u5bf9\u4e2d\u70b9\u53d6\u4e2d\u70b9，\u6700\u540e\u5f97\u5230\u66f2\u7ebf\u4e2d\u70b9\u3002\u5de6\u534a\u6bb5\u548c\u53f3\u534a\u6bb5\u5404\u81ea\u4ecd\u662f\u4e09\u6b21\u8d1d\u585e\u5c14，\u800c\u4e14\u63a7\u5236\u70b9\u53ef\u4ece\u8fd9\u5ea7\u4e09\u89d2\u5f62\u76f4\u63a5\u8bfb\u53d6\u3002\u76f8\u6bd4\u5c55\u5f00\u591a\u9879\u5f0f，De Casteljau \u6570\u503c\u7a33\u5b9a\u3001\u7ed3\u6784\u5bf9\u79f0，\u4e5f\u5f88\u9002\u5408\u9012\u5f52\u3002<\/p>\n

\u5b8c\u6574 C++17 \u4ee3\u7801<\/h3>\n

\u8f93\u51fa\u6298\u7ebf\u53ea\u5728\u5f00\u59cb\u653e\u5165 P0，\u6bcf\u4e2a\u53f6\u5b50\u6bb5\u8ffd\u52a0\u81ea\u5df1\u7684 P3，\u56e0\u6b64\u76f8\u90bb\u6bb5\u4e0d\u4f1a\u91cd\u590d\u4e2d\u70b9\u3002\u6df1\u5ea6\u4e0a\u9650\u9632\u6b62\u6781\u5c0f\u5bb9\u5dee\u6216\u5f02\u5e38\u6d6e\u70b9\u72b6\u6001\u9020\u6210\u65e0\u9650\u9012\u5f52\u3002<\/p>\n

#<\/span>include<\/span> <cassert><\/span><\/span>
\n#<\/span>include<\/span> <cmath><\/span><\/span>
\n#<\/span>include<\/span> <iostream><\/span><\/span>
\n#<\/span>include<\/span> <stdexcept><\/span><\/span>
\n#<\/span>include<\/span> <vector><\/span><\/span>
\nusing<\/span> namespace<\/span> std;<\/span>
\nstruct<\/span> Point<\/span> {<\/span> double<\/span> x,<\/span> y;<\/span> }<\/span>;<\/span>
\nPoint mid<\/span>(<\/span>Point a,<\/span> Point b)<\/span>{<\/span> return<\/span> {<\/span>(<\/span>a.<\/span>x+<\/span>b.<\/span>x)<\/span>\/<\/span>2<\/span>,<\/span> (<\/span>a.<\/span>y+<\/span>b.<\/span>y)<\/span>\/<\/span>2<\/span>}<\/span>;<\/span> }<\/span>
\ndouble<\/span> distanceToLine<\/span>(<\/span>Point p,<\/span> Point a,<\/span> Point b)<\/span>{<\/span>
\n double<\/span> dx=<\/span>b.<\/span>x–<\/span>a.<\/span>x,<\/span> dy=<\/span>b.<\/span>y–<\/span>a.<\/span>y;<\/span>
\n double<\/span> length=<\/span>hypot<\/span>(<\/span>dx,<\/span>dy)<\/span>;<\/span>
\n if<\/span>(<\/span>length==<\/span>0<\/span>)<\/span> return<\/span> hypot<\/span>(<\/span>p.<\/span>x–<\/span>a.<\/span>x,<\/span>p.<\/span>y–<\/span>a.<\/span>y)<\/span>;<\/span>
\n return<\/span> abs<\/span>(<\/span>dy*<\/span>p.<\/span>x–<\/span>dx*<\/span>p.<\/span>y+<\/span>b.<\/span>x*<\/span>a.<\/span>y–<\/span>b.<\/span>y*<\/span>a.<\/span>x)<\/span>\/<\/span>length;<\/span>
\n}<\/span>
\nvoid<\/span> flattenRec<\/span>(<\/span>Point p0,<\/span>Point p1,<\/span>Point p2,<\/span>Point p3,<\/span>double<\/span> tolerance,<\/span>int<\/span> depth,<\/span>vector<<\/span>Point><\/span>&<\/span> out)<\/span>{<\/span>
\n double<\/span> flat=<\/span>max<\/span>(<\/span>distanceToLine<\/span>(<\/span>p1,<\/span>p0,<\/span>p3)<\/span>,<\/span>distanceToLine<\/span>(<\/span>p2,<\/span>p0,<\/span>p3)<\/span>)<\/span>;<\/span>
\n if<\/span>(<\/span>flat<=<\/span>tolerance ||<\/span> depth==<\/span>24<\/span>)<\/span>{<\/span> out.<\/span>push_back<\/span>(<\/span>p3)<\/span>;<\/span> return<\/span>;<\/span> }<\/span>
\n Point q0=<\/span>mid<\/span>(<\/span>p0,<\/span>p1)<\/span>,<\/span> q1=<\/span>mid<\/span>(<\/span>p1,<\/span>p2)<\/span>,<\/span> q2=<\/span>mid<\/span>(<\/span>p2,<\/span>p3)<\/span>;<\/span>
\n Point r0=<\/span>mid<\/span>(<\/span>q0,<\/span>q1)<\/span>,<\/span> r1=<\/span>mid<\/span>(<\/span>q1,<\/span>q2)<\/span>,<\/span> s=<\/span>mid<\/span>(<\/span>r0,<\/span>r1)<\/span>;<\/span>
\n flattenRec<\/span>(<\/span>p0,<\/span>q0,<\/span>r0,<\/span>s,<\/span>tolerance,<\/span>depth+<\/span>1<\/span>,<\/span>out)<\/span>;<\/span>
\n flattenRec<\/span>(<\/span>s,<\/span>r1,<\/span>q2,<\/span>p3,<\/span>tolerance,<\/span>depth+<\/span>1<\/span>,<\/span>out)<\/span>;<\/span>
\n}<\/span>
\nvector<<\/span>Point><\/span> flatten<\/span>(<\/span>Point p0,<\/span>Point p1,<\/span>Point p2,<\/span>Point p3,<\/span>double<\/span> tolerance)<\/span>{<\/span>
\n if<\/span>(<\/span>!<\/span>(<\/span>tolerance><\/span>0<\/span>)<\/span>)<\/span> throw<\/span> invalid_argument<\/span>(<\/span>"positive tolerance required"<\/span>)<\/span>;<\/span>
\n vector<<\/span>Point><\/span> out{<\/span>p0}<\/span>;<\/span>
\n flattenRec<\/span>(<\/span>p0,<\/span>p1,<\/span>p2,<\/span>p3,<\/span>tolerance,<\/span>0<\/span>,<\/span>out)<\/span>;<\/span>
\n return<\/span> out;<\/span>
\n}<\/span>
\nint<\/span> main<\/span>(<\/span>)<\/span>{<\/span>
\n auto<\/span> line=<\/span>flatten<\/span>(<\/span>{<\/span>0<\/span>,<\/span>0<\/span>}<\/span>,<\/span>{<\/span>1<\/span>,<\/span>0<\/span>}<\/span>,<\/span>{<\/span>2<\/span>,<\/span>0<\/span>}<\/span>,<\/span>{<\/span>3<\/span>,<\/span>0<\/span>}<\/span>,<\/span>0.01<\/span>)<\/span>;<\/span>
\n assert<\/span>(<\/span>line.<\/span>size<\/span>(<\/span>)<\/span>==<\/span>2<\/span>)<\/span>;<\/span>
\n auto<\/span> curve=<\/span>flatten<\/span>(<\/span>{<\/span>0<\/span>,<\/span>0<\/span>}<\/span>,<\/span>{<\/span>0<\/span>,<\/span>3<\/span>}<\/span>,<\/span>{<\/span>3<\/span>,<\/span>3<\/span>}<\/span>,<\/span>{<\/span>3<\/span>,<\/span>0<\/span>}<\/span>,<\/span>0.1<\/span>)<\/span>;<\/span>
\n assert<\/span>(<\/span>curve.<\/span>size<\/span>(<\/span>)<\/span>><\/span>2<\/span>)<\/span>;<\/span>
\n assert<\/span>(<\/span>abs<\/span>(<\/span>curve.<\/span>front<\/span>(<\/span>)<\/span>.<\/span>x)<\/span><<\/span>1e-12<\/span> &&<\/span> abs<\/span>(<\/span>curve.<\/span>back<\/span>(<\/span>)<\/span>.<\/span>x–<\/span>3<\/span>)<\/span><<\/span>1e-12<\/span>)<\/span>;<\/span>
\n auto<\/span> finer=<\/span>flatten<\/span>(<\/span>{<\/span>0<\/span>,<\/span>0<\/span>}<\/span>,<\/span>{<\/span>0<\/span>,<\/span>3<\/span>}<\/span>,<\/span>{<\/span>3<\/span>,<\/span>3<\/span>}<\/span>,<\/span>{<\/span>3<\/span>,<\/span>0<\/span>}<\/span>,<\/span>0.02<\/span>)<\/span>;<\/span>
\n assert<\/span>(<\/span>finer.<\/span>size<\/span>(<\/span>)<\/span>>=<\/span>curve.<\/span>size<\/span>(<\/span>)<\/span>)<\/span>;<\/span>
\n cout <<<\/span> "coarse points: "<\/span> <<<\/span> curve.<\/span>size<\/span>(<\/span>)<\/span> <<<\/span> '\\\\n'<\/span>;<\/span>
\n cout <<<\/span> "fine points: "<\/span> <<<\/span> finer.<\/span>size<\/span>(<\/span>)<\/span> <<<\/span> '\\\\n'<\/span>;<\/span>
\n cout <<<\/span> "bezier tests passed\\\\n"<\/span>;<\/span>
\n}<\/span><\/p>\n

\u753b\u9762\u7b2c\u4e09\u5c42：\u9012\u5f52\u6811\u5982\u4f55\u53d8\u6210\u6298\u7ebf<\/h3>\n

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\u9000\u5316\u7aef\u70b9\u9700\u8981\u5355\u72ec\u7406\u89e3<\/h3>\n

\u5f53 P0==P3 \u65f6，\u5f26\u7ebf\u957f\u5ea6\u4e3a\u96f6，\u666e\u901a\u70b9\u7ebf\u8ddd\u79bb\u516c\u5f0f\u4f1a\u9664\u96f6\u3002\u4ee3\u7801\u9000\u5316\u4e3a\u63a7\u5236\u70b9\u5230\u7aef\u70b9\u7684\u6b27\u6c0f\u8ddd\u79bb，\u80fd\u7ee7\u7eed\u63a8\u52a8\u7ec6\u5206\u3002\u82e5\u56db\u4e2a\u63a7\u5236\u70b9\u5b8c\u5168\u76f8\u540c，\u5e73\u5766\u5ea6\u4e3a\u96f6，\u76f4\u63a5\u8f93\u51fa\u4e24\u4e2a\u76f8\u540c\u7aef\u70b9；\u8c03\u7528\u65b9\u53ef\u5728\u6298\u7ebf\u540e\u5904\u7406\u4e2d\u53bb\u91cd\u3002\u95ed\u5408\u4e0d\u7b49\u4e8e\u65e0\u66f2\u7ebf，\u5185\u90e8\u63a7\u5236\u70b9\u4ecd\u53ef\u80fd\u5f62\u6210\u73af\u72b6\u8def\u5f84\u3002<\/p>\n

\u4ece\u51e0\u4f55\u5bb9\u5dee\u5230\u5c4f\u5e55\u5bb9\u5dee<\/h3>\n

\u6a21\u578b\u5750\u6807\u4e2d\u7684 0.1 \u5728\u4e0d\u540c\u7f29\u653e\u4e0b\u4ee3\u8868\u4e0d\u540c\u50cf\u7d20\u8bef\u5dee\u3002\u6700\u7a33\u7684\u505a\u6cd5\u662f\u5728\u5e94\u7528\u6700\u7ec8\u53d8\u6362\u540e\u5224\u65ad\u5e73\u5766\u5ea6，\u6216\u628a\u50cf\u7d20\u5bb9\u5dee\u6309\u53d8\u6362\u6bd4\u4f8b\u6362\u56de\u6a21\u578b\u7a7a\u95f4\u3002\u975e\u5747\u5300\u7f29\u653e\u3001\u900f\u89c6\u53d8\u6362\u4e0b\u4e0d\u80fd\u53ea\u7528\u5355\u4e00\u6bd4\u4f8b\u8fd1\u4f3c\u3002\u82e5\u7528\u4e8e\u78b0\u649e\u68c0\u6d4b\u800c\u975e\u7ed8\u5236，\u8fd8\u8981\u8003\u8651\u6298\u7ebf\u4f4d\u4e8e\u66f2\u7ebf\u54ea\u4e00\u4fa7，\u7b80\u5355\u5e73\u5766\u5ea6\u4e0d\u81ea\u52a8\u7ed9\u51fa\u4fdd\u5b88\u5305\u56f4\u3002<\/p>\n

\u7528\u5305\u56f4\u76d2\u63d0\u524d\u6392\u9664\u4e0d\u53ef\u89c1\u66f2\u7ebf<\/h3>\n

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