{"id":95938,"date":"2026-08-27T03:48:40","date_gmt":"2026-08-26T19:48:40","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/95938.html"},"modified":"2026-08-27T03:48:40","modified_gmt":"2026-08-26T19:48:40","slug":"%e5%85%8b%e8%8e%b1%e5%9b%a0%e7%93%b6%e7%9a%84%e6%95%b0%e5%ad%a6%e5%8f%af%e8%a7%86%e5%8c%96%ef%bc%9a%e5%8e%9f%e7%90%86%e4%b8%8e%e5%ae%9e%e7%8e%b0","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/95938.html","title":{"rendered":"\u514b\u83b1\u56e0\u74f6\u7684\u6570\u5b66\u53ef\u89c6\u5316\uff1a\u539f\u7406\u4e0e\u5b9e\u73b0"},"content":{"rendered":"

import numpy as np<\/p>\n

self.play(
\n Create(cut_edge_line := Line(
\n start=np.array([-2, -0.5, -1]) + DOWN * 0.3,
\n end=np.array([-2, 0.5, -1]) + DOWN * 0.3,
\n color=YELLOW,
\n stroke_width=2,
\n )),
\n Write(cut_label),
\n run_time=1.5,
\n )
\n self.wait(1)<\/p>\n

# \u65cb\u8f6c\u5c55\u793a
\n self.begin_ambient_camera_rotation(rate=0.3)
\n self.wait(2)<\/p>\n

# 8. \u81ea\u52a8\u6f14\u793a\u62d3\u6251\u53d8\u6362（\u7b80\u5316\u7684\u52a8\u753b）
\n # \u521b\u5efa\u4e00\u4e2a\u5c0f\u7403\u4f53\u4ee3\u8868\u7a7a\u95f4\u70b9
\n point = Sphere(radius=0.1).set_color(RED)
\n point.move_to(klein_bottle_func(0, 0))
\n self.play(FadeIn(point))<\/p>\n

# \u8ba9\u70b9\u6cbf\u7740\u74f6\u9762\u79fb\u52a8
\n self.play(
\n UpdateFromAlphaFunc(
\n point,
\n lambda m, a: m.move_to(klein_bottle_func(2 * np.pi * a, 0))
\n ),
\n run_time=3,
\n )<\/p>\n

# 9. \u5faa\u73af\u5c55\u793a
\n self.play(
\n FadeOut(cut_edge_line),
\n FadeOut(cut_label),
\n FadeOut(point),
\n *[strip.animate.set_opacity(0.8) for strip in strips],
\n run_time=2,
\n )<\/p>\n

# \u5c55\u793a\u6700\u7ec8\u7ed3\u6784
\n self.play(
\n Rotate(strips, angle=2 * np.pi, axis=UP),
\n run_time=4,
\n rate_func=rate_funcs.smooth,
\n )<\/p>\n

self.begin_ambient_camera_rotation(rate=0.2)
\n self.wait(3)<\/p>\n

# \u505c\u6b62\u65cb\u8f6c\u5e76\u7b49\u5f85
\n self.stop_ambient_camera_rotation()
\n self.wait(1)<\/p>\n

# \u6de1\u51fa\u6240\u6709\u5143\u7d20
\n self.play(
\n *[FadeOut(mobj) for mobj in self.mobjects if mobj not in [title]],
\n title.animate.set_opacity(0),
\n run_time=2,
\n )
\n self.play(FadeOut(axes))<\/p>\n

# don't remove below command for run button to work
\n%manim -qm -v WARNING KleinBottleScene<\/p>\n

\u4e00\u3001\u514b\u83b1\u56e0\u74f6\u7684\u6570\u5b66\u6a21\u578b<\/h3>\n

1.1 \u514b\u83b1\u56e0\u74f6\u7684\u62d3\u6251\u5b9a\u4e49<\/h4>\n

\u514b\u83b1\u56e0\u74f6（Klein Bottle）\u662f\u5fb7\u56fd\u6570\u5b66\u5bb6\u83f2\u5229\u514b\u65af\u00b7\u514b\u83b1\u56e0（Felix Klein）\u4e8e1882\u5e74\u63d0\u51fa\u7684\u5355\u4fa7\u95ed\u66f2\u9762，\u5b83\u662f\u6700\u57fa\u672c\u7684\u4e0d\u53ef\u5b9a\u5411\u66f2\u9762\u4e4b\u4e00\u3002<\/p>\n

\u62d3\u6251\u5b66\u5b9a\u4e49：\u514b\u83b1\u56e0\u74f6\u662f\u7531\u4e00\u4e2a\u5706\u67f1\u4f53\u7ecf\u8fc7\u4ee5\u4e0b\u64cd\u4f5c\u6784\u9020\u800c\u6210：<\/p>\n

  • \u5c06\u5706\u67f1\u4f53\u7684\u4e00\u7aef\u626d\u8f6c180\u00b0\u540e\u4e0e\u53e6\u4e00\u7aef\u7c98\u5408<\/li>\n
  • \u6216\u8005\u7b49\u4ef7\u5730：\u5c06\u83ab\u6bd4\u4e4c\u65af\u5e26\u7684\u8fb9\u754c\u4e0e\u4e00\u4e2a\u5706\u76d8\u7c98\u5408<\/li>\n

    \u6838\u5fc3\u6027\u8d28：\u514b\u83b1\u56e0\u74f6\u5185\u90e8\u4e0e\u5916\u90e8\u8fde\u901a，\u6ca1\u6709\u660e\u786e\u7684\u5185\u5916\u4e4b\u5206（\u4e0d\u53ef\u5b9a\u5411\u6027）\u3002<\/p>\n

    1.2 \u53c2\u6570\u65b9\u7a0b<\/h4>\n

    \u514b\u83b1\u56e0\u74f6\u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u7684\u6807\u51c6\u53c2\u6570\u65b9\u7a0b\u4e3a：<\/p>\n

    \"\\\\left\\\\{\\\\begin{matrix}\"\"<\/p>\n

    \u5176\u4e2d：<\/p>\n

      \n
    • u \\\\in [0, 2\\\\pi)\\\\)：\u7ed5\u74f6\u4f53\u4e00\u5468\u7684\u89d2\u5ea6<\/li>\n
    • r > 0\\\\)：\u74f6\u8eab\u4e3b\u534a\u5f84（\u672c\u4ee3\u7801\u53d6 \\\\(r=2\\\\)）<\/li>\n<\/ul>\n

      \u8be5\u65b9\u7a0b\u5bf9\u5e94\u7684\u51e0\u4f55\u7ed3\u6784\u53ef\u5206\u89e3\u4e3a\u4e09\u4e2a\u9636\u6bb5：<\/p>\n

    • \u83ab\u6bd4\u4e4c\u65af\u5e26\u9636\u6bb5：\u5f53\u00a0v\u2208[0,2\u03c0]\u00a0\u65f6，\u5f62\u6210\u83ab\u6bd4\u4e4c\u65af\u5e26\u7684\u5c40\u90e8\u51e0\u4f55<\/li>\n
    • \u8fb9\u754c\u626d\u8f6c\u9636\u6bb5：\u5f53\u00a0v\u2208[2\u03c0,3\u03c0] \u65f6，\u66f2\u9762\u8fb9\u754c\u53d1\u751f\u534a\u626d\u8f6c<\/li>\n
    • \u95ed\u5408\u9636\u6bb5：\u5f53\u00a0v\u2208[3\u03c0,4\u03c0]\u65f6，\u66f2\u9762\u8fb9\u754c\u95ed\u5408，\u5f62\u6210\u81ea\u4ea4\u7684\u5c01\u95ed\u7ed3\u6784<\/li>\n
      \n

      \u4e8c\u3001\u4ee3\u7801\u5b9e\u73b0\u7684\u6570\u5b66\u539f\u7406<\/h3>\n

      2.1 \u53c2\u6570\u5316\u51fd\u6570\u8bbe\u8ba1<\/h4>\n

      def klein_bottle_func(u, v):
      \n """\u514b\u83b1\u56e0\u74f6\u53c2\u6570\u65b9\u7a0b"""
      \n x = (r + np.cos(u\/2)*np.sin(v) – np.sin(u\/2)*np.sin(2*v)) * np.cos(u)
      \n y = (r + np.cos(u\/2)*np.sin(v) – np.sin(u\/2)*np.sin(2*v)) * np.sin(u)
      \n z = np.sin(u\/2)*np.sin(v) + np.cos(u\/2)*np.sin(2*v)
      \n return np.array([x, y, z]) + UP*0.5<\/p>\n

      \u6570\u5b66\u5206\u6790：<\/p>\n

        \n
      • \u7cfb\u6570\u00a0r+cos\u2061(u\/2)sin\u2061v\u2212sin\u2061(u\/2)sin\u20612vr+cos(u\/2)sinv\u2212sin(u\/2)sin2v\u00a0\u51b3\u5b9a\u4e86\u74f6\u4f53\u7684\u5f84\u5411\u8ddd\u79bb<\/li>\n
      • \\\\cos(u\/2)\\\\) \u548c \\\\(\\\\sin(u\/2)\\\\) \u9879\u4ea7\u751f\u534a\u9891\u7387\u8c03\u5236，\u8fd9\u6b63\u662f\u83ab\u6bd4\u4e4c\u65af\u5e26\u626d\u8f6c\u7279\u6027\u7684\u4f53\u73b0<\/li>\n<\/ul>\n

        2.2 \u53c2\u6570\u57df\u5212\u5206<\/h4>\n\n\u53c2\u6570\u6570\u5b66\u7b26\u53f7\u8303\u56f4\u7269\u7406\u610f\u4e49<\/tr>\n\n\n\n
        u<\/td>\n\u5706\u5468\u89d2<\/td>\n[0,2\u03c0]<\/td>\n\u7ed5\u74f6\u4f53\u8f74\u7ebf\u65cb\u8f6c<\/td>\n<\/tr>\n
        v<\/td>\n\u6bcd\u7ebf\u53c2\u6570<\/td>\n[0,4\u03c0]<\/td>\n\u5f62\u6210\u53cc\u5c42\u7ed3\u6784<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

        \u91c7\u6837\u7b56\u7565：\u4ee3\u7801\u5c06\u00a0vv-\u8303\u56f4\u00a0[0,4\u03c0]\u5212\u5206\u4e3a7\u6bb5（\u6bcf\u6bb5\u5bbd\u4e3a\u00a0\"\\\\frac{4\u03c0\u200b}{7}\"\u200b），\u7528\u4e8e\u521b\u5efa7\u79cd\u4e0d\u540c\u989c\u8272\u7684\u8272\u5e26，\u589e\u5f3a\u62d3\u6251\u7ed3\u6784\u7684\u53ef\u89c6\u5316\u6548\u679c\u3002<\/p>\n

        \n

        \u4e09\u3001\u4ee3\u7801\u6a21\u5757\u529f\u80fd\u8be6\u89e3<\/h3>\n

        3.1 \u573a\u666f\u914d\u7f6e\u6a21\u5757<\/h4>\n

        class KleinBottleScene(ThreeDScene):
        \n def construct(self):
        \n self.camera.background_color = BLACKBOARD_GREEN
        \n self.set_camera_orientation(phi=70\u00b0, theta=-35\u00b0, zoom=0.85)<\/p>\n\n\u53c2\u6570\u4f5c\u7528<\/tr>\n\n\n\n\n\n
        background_color<\/td>\n\u8bbe\u7f6e\u573a\u666f\u80cc\u666f\u8272（\u9ed1\u7eff\u8272），\u6a21\u62df\u7269\u7406\u9ed1\u677f<\/td>\n<\/tr>\n
        phi=70\u00b0<\/td>\n\u76f8\u673a\u4fef\u4ef0\u89d2（\u76f8\u5bf9\u4e8e\u6c34\u5e73\u9762\u7684\u89d2\u5ea6）<\/td>\n<\/tr>\n
        theta=-35\u00b0<\/td>\n\u76f8\u673a\u6c34\u5e73\u65cb\u8f6c\u89d2（\u9006\u65f6\u9488\u4e3a\u8d1f）<\/td>\n<\/tr>\n
        zoom=0.85<\/td>\n\u76f8\u673a\u7f29\u653e\u7cfb\u6570（0.85\u500d）<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

        3.2 \u4e8c\u7ef4\u754c\u9762\u56fa\u5b9a\u6a21\u5757<\/h4>\n

        self.add_fixed_in_frame_mobjects(title, group1, group2, group3)<\/p>\n

        \u8be5\u8c03\u7528\u5c06\u56db\u4e2a\u4fe1\u606f\u7ec4\u56fa\u5b9a\u5728\u5c4f\u5e55\u5750\u6807\u7cfb\u4e0a，\u4e0d\u53d7\u4e09\u7ef4\u76f8\u673a\u65cb\u8f6c\u7684\u5f71\u54cd，\u786e\u4fdd\u6587\u5b57\u59cb\u7ec8\u9762\u5411\u89c2\u4f17\u3002<\/p>\n

        3.3 \u4e09\u7ef4\u5750\u6807\u8f74\u521b\u5efa\u6a21\u5757<\/h4>\n

        axes = ThreeDAxes(
        \n x_range=[-4, 4, 1], # x：\u4ece-4\u52304，\u523b\u5ea6\u95f4\u96941
        \n y_range=[-4, 4, 1], # y：\u4ece-4\u52304，\u523b\u5ea6\u95f4\u96941
        \n z_range=[-3, 3, 1], # z：\u4ece-3\u52303，\u523b\u5ea6\u95f4\u96941
        \n x_length=7, # \u663e\u793a\u957f\u5ea67\u5355\u4f4d
        \n y_length=7,
        \n z_length=5,
        \n)<\/p>\n

        \u989c\u8272\u6e10\u53d8\u8bbe\u7f6e：<\/p>\n

        for axis in [axes.x_axis, axes.y_axis, axes.z_axis]:
        \n axis.set_color(color=[RED, WHITE, RED])<\/p>\n

          \n
        • \u6bcf\u6761\u8f74\u4f7f\u7528\u7ea2-\u767d-\u7ea2\u6e10\u53d8\u7740\u8272<\/li>\n
        • \u6e10\u53d8\u8272\u6570\u7ec4\u5728Manim\u4e2d\u81ea\u52a8\u63d2\u503c<\/li>\n<\/ul>\n

          3.4 \u6587\u672c\u4e0e\u516c\u5f0f\u751f\u6210\u6a21\u5757<\/h4>\n\n\u5bf9\u8c61\u7c7b\u578b\u4f7f\u7528\u51fd\u6570\u7279\u70b9\u5e94\u7528\u573a\u666f<\/tr>\n\n\n\n\n
          Text()<\/td>\n\u7eaf\u6587\u672c<\/td>\n\u65e0\u6570\u5b66\u516c\u5f0f\u652f\u6301<\/td>\n\u6807\u9898\u3001\u6807\u7b7e<\/td>\n<\/tr>\n
          MathTex()<\/td>\nLaTeX\u516c\u5f0f<\/td>\n\u6570\u5b66\u6e32\u67d3<\/td>\n\u53c2\u6570\u65b9\u7a0b（eq1）<\/td>\n<\/tr>\n
          MarkupText()<\/td>\n\u5bcc\u6587\u672c<\/td>\n\u652f\u6301\\\\n\u6362\u884c<\/td>\n\u6027\u8d28\u6e05\u5355（eq2, eq3）<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

          \u793a\u4f8b\u8bf4\u660e：<\/p>\n

          eq1 = MathTex(
          \n r"x = \\\\cos u \\\\left( r + \\\\cos\\\\frac{u}{2} … \\\\right)\\\\\\\\", # \u7b2c\u4e00\u884c
          \n r"y = \\\\sin u …\\\\\\\\", # \u7b2c\u4e8c\u884c
          \n r"z = …", # \u7b2c\u4e09\u884c
          \n)<\/p>\n

            \n
          • MathTex\u00a0\u4e2d，\\\\\\\\\u00a0\u8868\u793a\u6362\u884c，\u4fdd\u6301\u7b49\u8ddd\u6392\u5217<\/li>\n
          • \\\\\\\\left(…\\\\\\\\right)\u00a0\u5b9e\u73b0\u81ea\u9002\u5e94\u7528\u62ec\u53f7<\/li>\n<\/ul>\n

            3.5 \u8868\u9762\u6784\u5efa\u6a21\u5757<\/h4>\n

            Simulated \u8868\u9762\u6784\u5efa\u4ee3\u7801\u5982\u4e0b（\u539f\u4ee3\u7801\u672a\u663e\u793a\u5b8c\u6574\u4f46\u57fa\u4e8eManim\u60ef\u4f8b）：<\/p>\n

            surface = Surface(
            \n lambda u, v: klein_bottle_func(u, v), # \u53c2\u6570\u65b9\u7a0b
            \n u_range=[0, 2*PI], # u\u53c2\u6570\u8303\u56f4
            \n v_range=[v_min, v_max], # v\u53c2\u6570\u8303\u56f4（\u5206\u6bb5）
            \n resolution=(32, 16), # \u7f51\u683c\u5206\u8fa8\u7387
            \n fill_opacity=0.7, # \u8868\u9762\u4e0d\u900f\u660e\u5ea6
            \n stroke_width=0.1, # \u7f51\u683c\u7ebf\u5bbd\u5ea6
            \n fill_color=color, # \u8868\u9762\u586b\u5145\u8272
            \n)<\/p>\n

            \u6838\u5fc3\u53c2\u6570\u89e3\u6790：<\/p>\n\n\u53c2\u6570\u542b\u4e49\u4f5c\u7528<\/tr>\n\n\n\n\n\n
            resolution=(32,16)<\/td>\n\u5728u\u65b9\u541132\u4e2a\u91c7\u6837\u70b9，v\u65b9\u541116\u4e2a\u91c7\u6837\u70b9<\/td>\n\u63a7\u5236\u7f51\u683c\u7ec6\u5bc6\u5ea6<\/td>\n<\/tr>\n
            fill_opacity=0.7<\/td>\n\u8868\u9762\u586b\u5145\u4e0d\u900f\u660e\u5ea6\u4e3a0.7<\/td>\n\u4f7f\u8868\u9762\u534a\u900f\u660e，\u53ef\u89c1\u5185\u90e8\u7ed3\u6784<\/td>\n<\/tr>\n
            stroke_width=0.1<\/td>\n\u7f51\u683c\u7ebf\u6761\u5bbd<\/td>\n\u663e\u793a\u7ebf\u6846\u67b6\u6784<\/td>\n<\/tr>\n
            fill_color<\/td>\n\u5355\u8272\u586b\u5145<\/td>\n7\u79cd\u989c\u8272\u4e4b\u4e00<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

            3.6 \u52a8\u753b\u521b\u5efa\u6a21\u5757<\/h4>\n

            \u63a8\u65ad\u52a8\u753b\u5e8f\u5217：<\/p>\n

            # 1. \u521b\u5efa\u5750\u6807\u8f74
            \nself.play(Create(axes), run_time=0.5)<\/p>\n

            # 2. \u9010\u6761\u663e\u793a\u8272\u5e26
            \nfor strip in strips:
            \n self.play(Create(strip), run_time=0.3)<\/p>\n

            # 3. \u76f8\u673a\u65cb\u8f6c
            \nself.begin_ambient_camera_rotation(rate=rate_funcs.linear)
            \nself.wait(3)<\/p>\n

              \n
            • Create(): \u663e\u793a\u5bf9\u8c61\u7684\u8f6e\u5ed3\u5e76\u9010\u6b65\u586b\u5145<\/li>\n
            • begin_ambient_camera_rotation(): \u81ea\u52a8\u65cb\u8f6c\u76f8\u673a，\u901f\u7387\u51fd\u6570\u4e3a\u7ebf\u6027(\u6052\u5b9a\u901f\u5ea6)<\/li>\n<\/ul>\n
              \n

              \u56db\u3001\u5173\u952e\u6570\u5b66\u7b97\u6cd5\u89e3\u91ca<\/h3>\n

              4.1 \u83ab\u6bd4\u4e4c\u65af\u5e26\u7684\u63a8\u5e7f<\/h4>\n

              \u514b\u83b1\u56e0\u74f6\u672c\u8d28\u4e0a\u53ef\u770b\u4f5c\u4e24\u4e2a\u83ab\u6bd4\u4e4c\u65af\u5e26\u7684\u8fb9\u754c\u7c98\u5408\u7684\u4ea7\u7269\u3002\u4ee3\u7801\u4e2d\u7684\u53c2\u6570\u5305\u542b：<\/p>\n

            • \u57fa\u7840\u73af\u90e8\u5206：(r+\u8c03\u5e45)\u22c5(cos\u2061u,sin\u2061u)\u5f62\u6210\u4e3b\u73af<\/li>\n
            • \u534a\u9891\u7387\u8c03\u5236：cos\u2061(u\/2)\u548c\u00a0sin\u2061(u\/2)\u4f7f\u6bcd\u7ebf\u626d\u8f6c180\u00b0，\u8fd9\u662f\u83ab\u6bd4\u4e4c\u65af\u5e26\u7684\u7279\u5f81<\/li>\n
            • \u53cc\u91cd\u5468\u671f：\u5728v\u65b9\u5411\u4f7f\u7528[0,4\u03c0]\u4f7f\u5f97\u8868\u9762\u672c\u8eab\u95ed\u5408（\u9700\u8f6c\u4e24\u5708\u624d\u56de\u5230\u8d77\u59cb\u4f4d\u7f6e）<\/li>\n

              4.2 \u81ea\u4ea4\u7ed3\u6784<\/h4>\n

              \u5f53\u00a0u=0 \u4e14\u00a0v=2\u03c0\u65f6，zz\u5206\u91cf\u9996\u5148\u5bf9\u00a0sin\u2061 \u9879\u4f7f\u8868\u9762\u81ea\u8eab\u7a7f\u900f，\u5f62\u6210\u6240\u8c13\u7684\u201c\u74f6\u9888\u201d\u81ea\u4ea4\u533a\u57df\u3002\u5728\u6570\u5b66\u4e0a，r=2\u00a0\u7684\u9009\u62e9\u907f\u514d\u4e86\u53c2\u6570\u7b49\u4ef7\u70b9\u9020\u6210\u7684\u9000\u5316\u3002<\/p>\n

              \n

              \u4e94\u3001\u4ee3\u7801\u4f18\u7f3a\u70b9\u5206\u6790<\/h3>\n

              \u4f18\u70b9<\/h4>\n
            • \u6a21\u5757\u5316\u8bbe\u8ba1：\u5c062D\u6587\u672c\u4e0e3D\u51e0\u4f55\u62d3\u6251\u5206\u79bb，\u4f7f\u7528add_fixed_in_frame\u89e3\u8026<\/li>\n
            • \u5f69\u8272\u5206\u533a：\u4e03\u79cd\u989c\u8272\u6e05\u6670\u5c55\u793a\u62d3\u6251\u7ed3\u6784<\/li>\n
            • \u6e10\u53d8\u8272\u5750\u6807\u8f74：\u89c6\u89c9\u7f8e\u89c2\u4e14\u76f4\u89c2<\/li>\n

              \u5f85\u6539\u8fdb\u5904<\/h4>\n
            • \u672a\u5b8c\u6210\u4ee3\u7801：\u539f\u4ee3\u7801\u5728strips\u521b\u5efa\u5faa\u73af\u540e\u672a\u5b8c\u6574\u663e\u793a，\u7f3a\u5c11Surface\u6784\u5efa\u90e8\u5206<\/li>\n
            • \u7f3a\u5c11\u52a8\u753b：self.play(…)\u5e8f\u5217\u672a\u5c55\u793a\u5b8c\u6574<\/li>\n
            • \u65e0\u4ea4\u4e92\u63a7\u5236：\u672a\u63d0\u4f9b\u7528\u6237\u8f93\u5165\u53c2\u6570\u8c03\u6574<\/li>\n
              \n

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