{"id":95936,"date":"2026-08-27T03:48:37","date_gmt":"2026-08-26T19:48:37","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/95936.html"},"modified":"2026-08-27T03:48:37","modified_gmt":"2026-08-26T19:48:37","slug":"%e8%8e%ab%e6%af%94%e4%b9%8c%e6%96%af%e5%b8%a6%e7%9a%84%e6%95%b0%e5%ad%a6%e5%8e%9f%e7%90%86%e5%92%8c%e8%a7%86%e8%a7%89%e7%89%b9%e6%80%a7","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/95936.html","title":{"rendered":"\u83ab\u6bd4\u4e4c\u65af\u5e26\u7684\u6570\u5b66\u539f\u7406\u548c\u89c6\u89c9\u7279\u6027"},"content":{"rendered":"

import numpy as np
\nfrom manim import *
\nimport manim.utils.rate_functions as rate_funcs<\/p>\n

# \u5b9a\u4e49\u9ed1\u677f\u7684\u9ed1\u7eff\u8272
\nBLACKBOARD_GREEN = "#1E352F"<\/p>\n

class MobiusRaceScene04(ThreeDScene):<\/p>\n

def construct(self):
\n # 1. \u8bbe\u7f6e\u9ed1\u677f\u80cc\u666f\u8272\u548c\u521d\u59cb\u76f8\u673a\u89c6\u89d2
\n self.camera.background_color = BLACKBOARD_GREEN
\n self.set_camera_orientation(
\n phi=70 * DEGREES, theta=-35 * DEGREES, zoom=0.85
\n )<\/p>\n

# —————- 2D \u6587\u672c\u4e0e\u516c\u5f0f (\u56fa\u5b9a\u5728\u5c4f\u5e55\u4e0a) —————-<\/p>\n

# \u6807\u9898 (\u9876\u90e8\u4e2d\u592e) – \u4f7f\u7528 Text \u4ee5\u5b8c\u7f8e\u652f\u6301\u4e2d\u6587
\n title = Text(
\n "\u83ab\u6bd4\u4e4c\u65af\u5e26\u7684\u6570\u5b66\u65b9\u7a0b\u4e0e\u5750\u6807\u53d8\u6362",
\n font="Sans",
\n color=YELLOW,
\n ).scale(0.7)
\n title.to_edge(UP, buff=0.4)<\/p>\n

# \u89d2\u843d (1): \u83ab\u6bd4\u4e4c\u65af\u5e26\u7684\u53c2\u6570\u65b9\u7a0b (\u5de6\u4e0a\u89d2)
\n label1 = Text("（1）\u83ab\u6bd4\u4e4c\u65af\u5e26\u53c2\u6570\u65b9\u7a0b：", font="Sans", color=WHITE).scale(0.35)
\n eq1 = MathTex(
\n r"x = \\\\left( R + v \\\\cos\\\\frac{u}{2} \\\\right) \\\\cos u\\\\\\\\",
\n r"y = \\\\left( R + v \\\\cos\\\\frac{u}{2} \\\\right) \\\\sin u\\\\\\\\",
\n r"z = v \\\\sin\\\\frac{u}{2}",
\n color=WHITE,
\n ).scale(0.55)
\n group1 = VGroup(label1, eq1).arrange(DOWN, aligned_edge=LEFT, buff=0.1)
\n group1.to_corner(UL).shift(DOWN * 0.8)<\/p>\n

# \u89d2\u843d (2): \u6781\u5750\u6807\u4e0e\u76f4\u89d2\u5750\u6807\u7684\u8f6c\u6362 (\u53f3\u4e0a\u89d2)
\n label2 = Text("（2）\u6781\u5750\u6807\u4e0e\u76f4\u89d2\u5750\u6807\u8f6c\u6362：", font="Sans", color=LIGHT_GRAY).scale(0.35)
\n eq2 = MathTex(
\n r"r = R + v \\\\cos\\\\frac{u}{2}\\\\\\\\",
\n r"x = r \\\\cos u\\\\\\\\",
\n r"y = r \\\\sin u",
\n color=LIGHT_GRAY,
\n ).scale(0.55)
\n group2 = VGroup(label2, eq2).arrange(DOWN, aligned_edge=LEFT, buff=0.1)
\n group2.to_corner(UR).shift(DOWN * 0.8)<\/p>\n

# \u89d2\u843d (3): \u7ed9\u5b9a\u6570\u5b57\u7684\u5177\u4f53\u8ba1\u7b97 (\u5de6\u4e0b\u89d2)
\n label3 = Text("（3）\u5b9e\u4f8b\u6570\u503c\u8ba1\u7b97：", font="Sans", color=ORANGE).scale(0.35)
\n eq3 = MathTex(
\n r"\\\\text{if} R = 2.5, u = \\\\pi, v = 0.5\\\\\\\\",
\n r"r = 2.5 + 0.5 \\\\cos(\\\\frac{\\\\pi}{2}) = 2.5\\\\\\\\",
\n r"x = 2.5 \\\\cos(\\\\pi) = -2.5\\\\\\\\",
\n r"y = 2.5 \\\\sin(\\\\pi) = 0\\\\\\\\",
\n r"z = 0.5 \\\\sin(\\\\frac{\\\\pi}{2}) = 0.5",
\n color=ORANGE,
\n ).scale(0.55)
\n group3 = VGroup(label3, eq3).arrange(DOWN, aligned_edge=LEFT, buff=0.1)
\n group3.to_corner(DL).shift(UP * 0.3)<\/p>\n

# \u4f7f\u7528 add_fixed_in_frame_mobjects \u786e\u4fdd 2D \u5143\u7d20\u4e0d\u968f 3D \u76f8\u673a\u65cb\u8f6c
\n self.add_fixed_in_frame_mobjects(title, group1, group2, group3)<\/p>\n

# —————- 3D \u51e0\u4f55\u4f53\u4e0e\u52a8\u753b —————-<\/p>\n

# \u521b\u5efa\u6e10\u53d8\u8272\u4e09\u7ef4\u5750\u6807\u8f74
\n axes = ThreeDAxes(
\n x_range=[-4, 4, 1],
\n y_range=[-4, 4, 1],
\n z_range=[-3, 3, 1],
\n x_length=7,
\n y_length=7,
\n z_length=5,
\n ).shift(
\n DOWN * 0.3
\n ) # \u7a0d\u5fae\u5411\u4e0b\u79fb\u52a8\u4ee5\u907f\u514d\u91cd\u53e0\u516c\u5f0f
\n for axis in [axes.x_axis, axes.y_axis, axes.z_axis]:
\n axis.set_color(color=[RED, WHITE, RED])<\/p>\n

# \u53c2\u6570\u5316\u83ab\u6bd4\u4e4c\u65af\u5e26\u51fd\u6570
\n R = 2.5<\/p>\n

def mobius_func(u, v):
\n x = (R + v * np.cos(u \/ 2)) * np.cos(u)
\n y = (R + v * np.cos(u \/ 2)) * np.sin(u)
\n z = v * np.sin(u \/ 2)
\n # \u540c\u6837\u5e94\u7528\u5e73\u79fb\u4ee5\u548c\u5750\u6807\u8f74\u5bf9\u9f50
\n return np.array([x, y, z]) + DOWN * 0.3<\/p>\n

# \u521b\u5efa\u4e03\u6761\u4e0d\u900f\u660e\u7684\u5e73\u884c\u8272\u5e26
\n colors = [RED, ORANGE, YELLOW, GREEN, TEAL, BLUE, PURPLE]
\n num_strips = 7
\n total_width = 1.0
\n strip_width = total_width \/ num_strips<\/p>\n

strips = VGroup()
\n for i, color in enumerate(colors):
\n v_min = -total_width \/ 2 + i * strip_width
\n v_max = v_min + strip_width<\/p>\n

strip = Surface(
\n lambda u, v, v_min=v_min, v_max=v_max: mobius_func(u, v),
\n u_range=[0, 2 * np.pi],
\n v_range=[v_min, v_max],
\n resolution=(60, 2),
\n )
\n strip.set_fill(color=color, opacity=1.0)
\n strip.set_stroke(width=0)
\n strips.add(strip)<\/p>\n

# \u5b9a\u4e49 3 \u4e2a\u201c\u8682\u8681\u201d (\u9002\u914d v0.19.0)
\n ant_sphere = Sphere(radius=0.15).set_color(RED)
\n ant_cube = Cube(side_length=0.25).set_color(GREEN)
\n ant_tetra = Tetrahedron(
\n edge_length=0.3,
\n faces_config={"fill_color": BLUE, "fill_opacity": 1.0},
\n )
\n ant_tetra.graph.set_color(BLUE)<\/p>\n

ants = [ant_sphere, ant_cube, ant_tetra]<\/p>\n

def get_ant_position(alpha, index):
\n base_u = alpha * 4 * np.pi
\n offset = index * (4 * np.pi \/ 3)
\n u = (base_u + offset) % (4 * np.pi)
\n return mobius_func(u, 0)<\/p>\n

# —————- \u52a8\u753b\u64ad\u653e —————-<\/p>\n

# 1. \u6e10\u663e\u6807\u9898\u548c\u516c\u5f0f\u7ec4
\n self.play(
\n Write(title),
\n FadeIn(group1, shift=RIGHT),
\n FadeIn(group2, shift=LEFT),
\n FadeIn(group3, shift=UP),
\n run_time=2,
\n )
\n self.wait(0.5)<\/p>\n

# 2. \u7ed8\u5236\u5750\u6807\u8f74
\n self.play(Create(axes), run_time=2)
\n self.wait(0.5)<\/p>\n

# 3. \u8272\u5e26\u8d5b\u8dd1
\n rate_functions = [
\n rate_funcs.smooth,
\n rate_funcs.ease_in_quad,
\n rate_funcs.ease_out_quad,
\n rate_funcs.linear,
\n rate_funcs.ease_in_out_cubic,
\n rate_funcs.ease_in_sine,
\n rate_funcs.ease_out_sine,
\n ]<\/p>\n

strip_animations = []
\n for i, strip in enumerate(strips):
\n strip_animations.append(
\n Create(
\n strip,
\n rate_func=rate_functions[i],
\n run_time=6,
\n )
\n )<\/p>\n

self.play(*strip_animations)
\n self.wait(0.5)<\/p>\n

# 4. \u76f8\u673a\u4e0e\u573a\u666f\u65cb\u8f6c (\u6b64\u65f6\u5c4f\u5e55\u89d2\u843d\u7684\u516c\u5f0f\u4fdd\u6301\u4e0d\u52a8)
\n self.begin_ambient_camera_rotation(rate=0.4)
\n self.play(
\n Rotate(axes, angle=2 * np.pi, axis=OUT),
\n Rotate(strips, angle=2 * np.pi, axis=OUT),
\n run_time=4,
\n rate_func=rate_funcs.linear,
\n )
\n self.stop_ambient_camera_rotation()
\n self.wait(0.5)<\/p>\n

# 5. \u8682\u8681\u722c\u884c
\n for i, ant in enumerate(ants):
\n ant.move_to(get_ant_position(0, i))
\n self.play(*[FadeIn(ant) for ant in ants])<\/p>\n

def update_ant(ant, alpha, index):
\n ant.move_to(get_ant_position(alpha, index))
\n ant.rotate(
\n 0.15, axis=np.array([1, 1, 1]), about_point=ant.get_center()
\n )<\/p>\n

self.play(
\n UpdateFromAlphaFunc(
\n ant_sphere,
\n lambda m, a: update_ant(m, a, 0),
\n ),
\n UpdateFromAlphaFunc(
\n ant_cube,
\n lambda m, a: update_ant(m, a, 1),
\n ),
\n UpdateFromAlphaFunc(
\n ant_tetra,
\n lambda m, a: update_ant(m, a, 2),
\n ),
\n run_time=10,
\n rate_func=rate_funcs.linear,
\n )<\/p>\n

self.wait(2)<\/p>\n

\u8fd9\u6bb5\u4ee3\u7801\u4f7f\u7528Manim\u5e93\u521b\u5efa\u4e86\u4e00\u4e2a3D\u52a8\u753b，\u5c55\u793a\u4e86\u83ab\u6bd4\u4e4c\u65af\u5e26\u7684\u6570\u5b66\u539f\u7406\u548c\u89c6\u89c9\u7279\u6027\u3002\u8ba9\u6211\u8be6\u7ec6\u89e3\u91ca\u4ee3\u7801\u7684\u5404\u4e2a\u90e8\u5206：<\/p>\n

\u4e3b\u8981\u529f\u80fd\u6982\u8ff0<\/h3>\n

\u8fd9\u662f\u4e00\u4e2a\u5173\u4e8e\u83ab\u6bd4\u4e4c\u65af\u5e26（Mobius strip）\u76843D\u6570\u5b66\u6f14\u793a\u52a8\u753b，\u5305\u542b：<\/p>\n

  • \u6570\u5b66\u516c\u5f0f\u5c55\u793a<\/li>\n
  • 3D\u51e0\u4f55\u4f53\u6784\u5efa<\/li>\n
  • \u591a\u5143\u7d20\u52a8\u753b\u540c\u6b65<\/li>\n
  • \u76f8\u673a\u8fd0\u52a8\u63a7\u5236<\/li>\n

    \u4ee3\u7801\u7ed3\u6784\u8be6\u89e3<\/h3>\n

    1. \u573a\u666f\u8bbe\u7f6e<\/h4>\n

    self.camera.background_color = BLACKBOARD_GREEN
    \nself.set_camera_orientation(phi=70 * DEGREES, theta=-35 * DEGREES, zoom=0.85)<\/p>\n

      \n
    • \u8bbe\u7f6e\u9ed1\u7eff\u8272\u80cc\u666f（\u9ed1\u677f\u98ce\u683c）<\/li>\n
    • \u8c03\u65743D\u76f8\u673a\u521d\u59cb\u89d2\u5ea6\u548c\u7f29\u653e<\/li>\n<\/ul>\n

      2. 2D\u6587\u672c\u4e0e\u516c\u5f0f（\u56fa\u5b9a\u5728\u5c4f\u5e55）<\/h4>\n

      \u4f7f\u7528add_fixed_in_frame_mobjects\u57283D\u573a\u666f\u4e2d\u56fa\u5b9a2D\u5143\u7d20：<\/p>\n

        \n
      • \u5de6\u4e0a：\u83ab\u6bd4\u4e4c\u65af\u5e26\u53c2\u6570\u65b9\u7a0b<\/li>\n
      • \u53f3\u4e0a：\u6781\u5750\u6807\u4e0e\u76f4\u89d2\u5750\u6807\u8f6c\u6362<\/li>\n
      • \u5de6\u4e0b：\u5177\u4f53\u6570\u503c\u8ba1\u7b97\u793a\u4f8b<\/li>\n
      • \u9876\u90e8：\u6807\u9898<\/li>\n<\/ul>\n

        \u5373\u4f7f3D\u573a\u666f\u65cb\u8f6c，\u8fd9\u4e9b2D\u5143\u7d20\u59cb\u7ec8\u9762\u5411\u6444\u50cf\u673a\u3002<\/p>\n

        3. \u83ab\u6bd4\u4e4c\u65af\u5e26\u76843D\u6784\u5efa<\/h4>\n

        def mobius_func(u, v):
        \n x = (R + v * np.cos(u \/ 2)) * np.cos(u)
        \n y = (R + v * np.cos(u \/ 2)) * np.sin(u)
        \n z = v * np.sin(u \/ 2)<\/p>\n

        \u8fd9\u662f\u83ab\u6bd4\u4e4c\u65af\u5e26\u7684\u6807\u51c6\u53c2\u6570\u65b9\u7a0b，\u5176\u4e2d：<\/p>\n

          \n
        • u：\u7ed5\u73af\u89d2\u5ea6（0\u52302\u03c0）<\/li>\n
        • v：\u5e26\u5bbd\u5ea6\u65b9\u5411（-0.5\u52300.5）<\/li>\n
        • \u901a\u8fc7v * cos(u\/2)\u548cv * sin(u\/2)\u5b9e\u73b0\u534a\u626d\u8f6c<\/li>\n<\/ul>\n

          4. \u4e03\u8272\u5e26\u6784\u5efa<\/h4>\n

          \u5c06\u83ab\u6bd4\u4e4c\u65af\u5e26\u5206\u5272\u62107\u4e2a\u4e0d\u540c\u989c\u8272\u7684\u5e73\u884c\u6761\u5e26，\u6bcf\u4e2a\u6761\u5e26\u4f7f\u7528\u4e0d\u540c\u7684\u52a8\u753b\u901f\u7387\u51fd\u6570\u3002<\/p>\n

          5. \u52a8\u753b\u987a\u5e8f<\/h4>\n
        • \u6807\u9898\u548c\u516c\u5f0f\u6e10\u663e（2\u79d2）<\/li>\n
        • \u7ed8\u5236\u5750\u6807\u8f74（2\u79d2）<\/li>\n
        • \u6761\u5e26\u52a8\u753b（6\u79d2）- \u4f7f\u7528\u4e0d\u540c\u901f\u7387\u51fd\u6570<\/li>\n
        • \u573a\u666f\u65cb\u8f6c（4\u79d2）：\u76f8\u673a\u81ea\u52a8\u65cb\u8f6c+\u573a\u666f\u6574\u4f53\u65cb\u8f6c<\/li>\n
        • \u8682\u8681\u52a8\u753b（10\u79d2）：3\u53ea"\u8682\u8681"\u5728\u5e26\u4e0a\u884c\u8d70<\/li>\n

          6. \u8682\u8681\u52a8\u753b<\/h4>\n
            \n
          • \u7ea2\u7403（Sphere）<\/li>\n
          • \u7eff\u65b9\u5757（Cube）<\/li>\n
          • \u84dd\u56db\u9762\u4f53（Tetrahedron）<\/li>\n<\/ul>\n

            \u901a\u8fc7UpdateFromAlphaFunc\u66f4\u65b0\u4f4d\u7f6e，\u4f7f\u7528\u56fa\u5b9a\u504f\u79fb\u5b9e\u73b0\u7b49\u95f4\u8ddd\u5206\u5e03\u3002<\/p>\n

            \u5173\u952e\u6280\u672f\u70b9<\/h3>\n

            \u4e0d\u540c\u901f\u7387\u51fd\u6570<\/h4>\n

            rate_functions = [
            \n rate_funcs.smooth, # \u5e73\u6ed1
            \n rate_funcs.ease_in_quad, # \u4e8c\u6b21\u7f13\u5165
            \n rate_funcs.ease_out_quad, # \u4e8c\u6b21\u7f13\u51fa
            \n rate_funcs.linear, # \u7ebf\u6027
            \n rate_funcs.ease_in_out_cubic, # \u4e09\u6b21\u7f13\u5165\u7f13\u51fa
            \n rate_funcs.ease_in_sine, # \u6b63\u5f26\u7f13\u5165
            \n rate_funcs.ease_out_sine, # \u6b63\u5f26\u7f13\u51fa
            \n]<\/p>\n

            \u6bcf\u4e2a\u6761\u5e26\u4f7f\u7528\u4e0d\u540c\u901f\u5ea6\u66f2\u7ebf，\u5c55\u793a\u76f8\u540c\u51e0\u4f55\u4f53\u5728\u4e0d\u540c\u8fd0\u52a8\u5b66\u4e0b\u7684\u6548\u679c\u3002<\/p>\n

            3D\u52302D\u7684\u8f6c\u6362<\/h4>\n
              \n
            • \u4f7f\u7528add_fixed_in_frame_mobjects\u56fa\u5b9a\u5c4f\u5e55\u5143\u7d20<\/li>\n
            • 3D\u5143\u7d20\u6b63\u5e38\u53c2\u4e0e\u7a7a\u95f4\u53d8\u6362<\/li>\n
            • \u76f8\u673a\u89d2\u5ea6\u53d8\u5316\u4e0d\u5f71\u54cd\u56fa\u5b9a\u76842D\u5143\u7d20<\/li>\n<\/ul>\n

              \u53c2\u6570\u65b9\u7a0b\u7684\u5e94\u7528<\/h4>\n
                \n
              • \u4ee3\u7801\u540c\u65f6\u5c55\u793a\u4e86\u6570\u5b66\u7406\u8bba\u548c3D\u5b9e\u73b0<\/li>\n
              • \u6240\u67093D\u51e0\u4f55\u4f53\u7531\u53c2\u6570\u65b9\u7a0b\u751f\u6210<\/li>\n
              • \u901a\u8fc7\u6539\u53d8\u53c2\u6570\u4ea7\u751f\u52a8\u753b\u6548\u679c<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"

                import numpy as np
                \nfrom manim import *
                \nimport manim.utils.rate_functions as rate_funcs# \u5b9a\u4e49\u9ed1\u677f\u7684\u9ed1\u7eff\u8272
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