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MatheMatica 绘制鸟巢体育馆

交互·渲染|动画|字数 220|阅读时长≈ 1 分钟

背景

fig.1 A 3D graph projected onto the xy-plane
fig.1 A 3D graph projected onto the xy-plane

YouTube 上刷到一个用 manim 做的实验动画——3D 图形投影到 xy 平面，效果挺有意思。其数学原理是使用三角函数绘制凸凹的椭圆曲线，最近在学 MatheMatica，拿它来练练手。不过光是照搬太单调，不如发挥一下，把它变成一个“鸟巢”造型。

fig.2 鸟巢图（来源：光明网）
fig.2 鸟巢图（来源：光明网）

绘制流程

最好先搞清几个关键形状的数学原理，比如中间内凹的椭圆、穿过四个点的曲线等。

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mathematica 绘制中间内凹的椭圆

椭圆绘制基于参数方程 r(t)=( 2 cos(t), 2 sin(t), 0.5 sin(2 t) + 2 ) ，通过调整参数即可得到不同的形态。 该椭圆由参数方程 r(t)=(2cos⁡t,2sin⁡t,0.5sin⁡(2t)+2)r(t)=(2cost,2sint,0.5sin(2t)+2) 通过调整参数生成。

Code
curve = ParametricPlot3D[{4 Cos[t], 4.2 Sin[t],     0.6 Sin[2 t + Pi + Pi/2] + 3.2}, {t, 0, 2 Pi},    PlotStyle -> {Thick, Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}]; axes = Graphics3D[{Arrowheads[0.05], {Red,      Arrow[{{0, 0, 0}, {5, 0, 0}}]},(*X 轴*){Green,      Arrow[{{0, 0, 0}, {0, 5, 0}}]},(*Y 轴*){Blue,      Arrow[{{0, 0, 0}, {0, 0, 5}}]}    (*Z 轴*)}];  png = Show[curve, axes, AxesLabel -> {"X", "Y", "Z"},   LabelStyle -> {FontSize -> 14}, Boxed -> True]

mathematica 绘制鸟巢几条主要的轮廓线
Code
curve12 =   ParametricPlot3D[{2 Cos[t], 1.5 Sin[t],     0.2 Sin[2 t + Pi + Pi/2] + 3}, {t, 0, 2 Pi},    PlotStyle -> {Thick, Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];curve21 =   ParametricPlot3D[{4 Cos[t], 4.2 Sin[t],     0.6 Sin[2 t + Pi + Pi/2] + 3.2}, {t, 0, 2 Pi},    PlotStyle -> {Thick, Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];curve22 =   ParametricPlot3D[{4.2 Cos[t], 4.4 Sin[t],     0.6 Sin[2 t + Pi + Pi/2] + 2.5}, {t, 0, 2 Pi},    PlotStyle -> {Thick, Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];curve31 =   ParametricPlot3D[{3.4 Cos[t], 3.8 Sin[t], 0}, {t, 0, 2 \[Pi]},    PlotStyle -> {Thick, Blue}, Axes -> False,    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}]; axes = Graphics3D[{Arrowheads[0.05], {Red,      Arrow[{{0, 0, 0}, {5, 0, 0}}]},(*X 轴*){Green,      Arrow[{{0, 0, 0}, {0, 5, 0}}]},(*Y 轴*){Blue,      Arrow[{{0, 0, 0}, {0, 0, 5}}]}    (*Z 轴*)}];  png = Show[curve12, curve21, curve22, curve31, axes,   AxesLabel -> {"X", "Y", "Z"}, LabelStyle -> {FontSize -> 14},   Boxed -> True]
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mathematica 四段曲线上分别选取一个点，连接成线
Code
curve12 =   ParametricPlot3D[{2 Cos[t], 1.5 Sin[t],     0.2 Sin[2 t + Pi + Pi/2] + 3}, {t, 0, 2 Pi},    PlotStyle -> {Thick, Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}]; curve21 =   ParametricPlot3D[{4 Cos[t], 4.2 Sin[t],     0.6 Sin[2 t + Pi + Pi/2] + 3.2}, {t, 0, 2 Pi},    PlotStyle -> {Thick, Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}]; curve22 =   ParametricPlot3D[{4.2 Cos[t], 4.4 Sin[t],     0.6 Sin[2 t + Pi + Pi/2] + 2.5}, {t, 0, 2 Pi},    PlotStyle -> {Thick, Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}]; curve31 =   ParametricPlot3D[{3.4 Cos[t], 3.8 Sin[t], 0}, {t, 0, 2 \[Pi]},    PlotStyle -> {Thick, Blue}, Axes -> False,    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}]; axes = Graphics3D[{Arrowheads[0.05], {Red,      Arrow[{{0, 0, 0}, {5, 0, 0}}]},(*X 轴*){Green,      Arrow[{{0, 0, 0}, {0, 5, 0}}]},(*Y 轴*){Blue,      Arrow[{{0, 0, 0}, {0, 0, 5}}]}    (*Z 轴*)}];  p1 = {2 Cos[0], 1.5 Sin[0],    0.2 Sin[2*0 + Pi + Pi/2] + 3};       (*curve12 上的点*)p2 = {4 Cos[0], 4.2 Sin[0],    0.6 Sin[2*0 + Pi + Pi/2] + 3.2};       (*curve21 上的点*)p3 = {4.2 Cos[0], 4.4 Sin[0],    0.6 Sin[2*0 + Pi + Pi/2] + 2.5};   (*curve22 上的点*)p4 = {3.4 Cos[0], 3.8 Sin[0],    0};                                (*curve31 上的点*) (*连接四个点绘制红色直线*)connector =   Graphics3D[{Thick, Red, Line[{p1, p2, p3, p4}],(*连接四个点的折线*)    PointSize[0.02], {Black, Point[p1]},(*点1：黑色*){Black,      Point[p2]},(*点2：黑色*){Black, Point[p3]},(*点3：黑色*){Black,      Point[p4]}                 (*点4：黑色*)}];  png = Show[curve12, curve21, curve22, curve31, axes, connector,   AxesLabel -> {"X", "Y", "Z"}, LabelStyle -> {FontSize -> 14},   Boxed -> True]
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mathematica 四段曲线上分别选取一个点，直线替换为曲线
Code
curve12 =   ParametricPlot3D[{2 Cos[t], 1.5 Sin[t],     0.2 Sin[2 t + Pi + Pi/2] + 3}, {t, 0, 2 Pi},    PlotStyle -> {Thick, Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];curve21 =   ParametricPlot3D[{4 Cos[t], 4.2 Sin[t],     0.6 Sin[2 t + Pi + Pi/2] + 3.2}, {t, 0, 2 Pi},    PlotStyle -> {Thick, Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];curve22 =   ParametricPlot3D[{4.2 Cos[t], 4.4 Sin[t],     0.6 Sin[2 t + Pi + Pi/2] + 2.5}, {t, 0, 2 Pi},    PlotStyle -> {Thick, Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];curve31 =   ParametricPlot3D[{3.4 Cos[t], 3.8 Sin[t], 0}, {t, 0, 2 \[Pi]},    PlotStyle -> {Thick, Blue}, Axes -> False,    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}]; axes = Graphics3D[{Arrowheads[0.05], {Red,      Arrow[{{0, 0, 0}, {5, 0, 0}}]},(*X 轴*){Green,      Arrow[{{0, 0, 0}, {0, 5, 0}}]},(*Y 轴*){Blue,      Arrow[{{0, 0, 0}, {0, 0, 5}}]}    (*Z 轴*)}];  p1 = {2 Cos[0], 1.5 Sin[0],    0.2 Sin[2*0 + Pi + Pi/2] + 3};       (*curve12 上的点*)p2 = {4 Cos[0], 4.2 Sin[0],    0.6 Sin[2*0 + Pi + Pi/2] + 3.2};       (*curve21 上的点*)p3 = {4.2 Cos[0], 4.4 Sin[0],    0.6 Sin[2*0 + Pi + Pi/2] + 2.5};   (*curve22 上的点*)p4 = {3.4 Cos[0], 3.8 Sin[0],    0};                                (*curve31 上的点*)  catmullRom3D[{p0_, p1_, p2_, p3_}, t_] :=   0.5*{2 p1[[1]] + t (p2[[1]] - p0[[1]]) +      t^2 (2 p0[[1]] - 5 p1[[1]] + 4 p2[[1]] - p3[[1]]) +      t^3 (-p0[[1]] + 3 p1[[1]] - 3 p2[[1]] + p3[[1]]),     2 p1[[2]] + t (p2[[2]] - p0[[2]]) +      t^2 (2 p0[[2]] - 5 p1[[2]] + 4 p2[[2]] - p3[[2]]) +      t^3 (-p0[[2]] + 3 p1[[2]] - 3 p2[[2]] + p3[[2]]),     2 p1[[3]] + t (p2[[3]] - p0[[3]]) +      t^2 (2 p0[[3]] - 5 p1[[3]] + 4 p2[[3]] - p3[[3]]) +      t^3 (-p0[[3]] + 3 p1[[3]] - 3 p2[[3]] + p3[[3]])}; pts = {p1, p2, p3, p4};newPts =   Join[{2.1 pts[[1]] - pts[[2]]}, pts, {2.2 pts[[-1]] - pts[[-2]]}]; curvePoints3D =   Table[catmullRom3D[{p1, p2, p3, p4}, t], {t, 0, 1, 0.1}]; curve = Graphics3D[{{Thick, Blue, Line[curvePoints3D]}, {Black,      PointSize[0.02], Point[{p1, p1, p2, p3}]}, {Dashed, Gray,      Line[{p1, p1, p2, p3}]}}];  png = Show[curve, curve12, curve21, curve22, curve31, axes,   AxesLabel -> {"X", "Y", "Z"}, LabelStyle -> {FontSize -> 14},   Boxed -> True]
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mathematica 四段曲线上分别选取一个点，添加两个辅助点，直线替换为曲线
Code
curve12 =   ParametricPlot3D[{2 Cos[t], 1.5 Sin[t],     0.2 Sin[2 t + Pi + Pi/2] + 3}, {t, 0, 2 Pi},    PlotStyle -> {Thick, Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];curve21 =   ParametricPlot3D[{4 Cos[t], 4.2 Sin[t],     0.6 Sin[2 t + Pi + Pi/2] + 3.2}, {t, 0, 2 Pi},    PlotStyle -> {Thick, Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];curve22 =   ParametricPlot3D[{4.2 Cos[t], 4.4 Sin[t],     0.6 Sin[2 t + Pi + Pi/2] + 2.5}, {t, 0, 2 Pi},    PlotStyle -> {Thick, Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];curve31 =   ParametricPlot3D[{3.4 Cos[t], 3.8 Sin[t], 0}, {t, 0, 2 \[Pi]},    PlotStyle -> {Thick, Blue}, Axes -> False,    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}]; axes = Graphics3D[{Arrowheads[0.05], {Red,      Arrow[{{0, 0, 0}, {5, 0, 0}}]},(*X 轴*){Green,      Arrow[{{0, 0, 0}, {0, 5, 0}}]},(*Y 轴*){Blue,      Arrow[{{0, 0, 0}, {0, 0, 5}}]}    (*Z 轴*)}]; p1 = {2 Cos[0], 1.5 Sin[0],    0.2 Sin[2*0 + Pi + Pi/2] + 3};       (*curve12 上的点*)p2 = {4 Cos[0], 4.2 Sin[0],    0.6 Sin[2*0 + Pi + Pi/2] + 3.2};       (*curve21 上的点*)p3 = {4.2 Cos[0], 4.4 Sin[0],    0.6 Sin[2*0 + Pi + Pi/2] + 2.5};   (*curve22 上的点*)p4 = {3.4 Cos[0], 3.8 Sin[0],    0};                                (*curve31 上的点*) catmullRom3D[{p0_, p1_, p2_, p3_}, t_] :=   0.5*{2 p1[[1]] + t (p2[[1]] - p0[[1]]) +      t^2 (2 p0[[1]] - 5 p1[[1]] + 4 p2[[1]] - p3[[1]]) +      t^3 (-p0[[1]] + 3 p1[[1]] - 3 p2[[1]] + p3[[1]]),     2 p1[[2]] + t (p2[[2]] - p0[[2]]) +      t^2 (2 p0[[2]] - 5 p1[[2]] + 4 p2[[2]] - p3[[2]]) +      t^3 (-p0[[2]] + 3 p1[[2]] - 3 p2[[2]] + p3[[2]]),     2 p1[[3]] + t (p2[[3]] - p0[[3]]) +      t^2 (2 p0[[3]] - 5 p1[[3]] + 4 p2[[3]] - p3[[3]]) +      t^3 (-p0[[3]] + 3 p1[[3]] - 3 p2[[3]] + p3[[3]])}; pts = {p1, p2, p3, p4};curvePoints3D =   Table[catmullRom3D[{p1, p2, p3, p4}, t], {t, 0, 1, 0.1}]; newPts =   Join[{2.1 pts[[1]] - pts[[2]]}, pts, {2.2 pts[[-1]] - pts[[-2]]}];crPts = Flatten[   Table[Table[     catmullRom3D[newPts[[i ;; i + 3]], s], {s, 0, 1, 0.1}], {i, 1,      Length[newPts] - 3}], 1]; curve = Graphics3D[{{Thick, Blue, Line[crPts]}, {Black,      PointSize[0.02], Point[{p1, p1, p2, p3}]}, {Dashed, Gray,      Line[{p1, p1, p2, p3}]}}]; png = Show[curve, curve12, curve21, curve22, curve31, axes,   AxesLabel -> {"X", "Y", "Z"}, LabelStyle -> {FontSize -> 14},   Boxed -> True]
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mathematica 绘制一圈曲线
Code
curve12 =   ParametricPlot3D[{2 Cos[t], 1.5 Sin[t],     0.2 Sin[2 t + Pi + Pi/2] + 3}, {t, 0, 2 Pi},    PlotStyle -> {Thick, Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];curve21 =   ParametricPlot3D[{4 Cos[t], 4.2 Sin[t],     0.6 Sin[2 t + Pi + Pi/2] + 3.2}, {t, 0, 2 Pi},    PlotStyle -> {Thick, Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];curve22 =   ParametricPlot3D[{4.2 Cos[t], 4.4 Sin[t],     0.6 Sin[2 t + Pi + Pi/2] + 2.5}, {t, 0, 2 Pi},    PlotStyle -> {Thick, Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];curve31 =   ParametricPlot3D[{3.4 Cos[t], 3.8 Sin[t], 0}, {t, 0, 2 \[Pi]},    PlotStyle -> {Thick, Blue}, Axes -> False,    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}]; axes = Graphics3D[{Arrowheads[0.05], {Red,      Arrow[{{0, 0, 0}, {5, 0, 0}}]},(*X 轴*){Green,      Arrow[{{0, 0, 0}, {0, 5, 0}}]},(*Y 轴*){Blue,      Arrow[{{0, 0, 0}, {0, 0, 5}}]}    (*Z 轴*)}]; catmullRom3D[{p0_, p1_, p2_, p3_}, t_] :=   0.5*{2 p1[[1]] + t (p2[[1]] - p0[[1]]) +      t^2 (2 p0[[1]] - 5 p1[[1]] + 4 p2[[1]] - p3[[1]]) +      t^3 (-p0[[1]] + 3 p1[[1]] - 3 p2[[1]] + p3[[1]]),     2 p1[[2]] + t (p2[[2]] - p0[[2]]) +      t^2 (2 p0[[2]] - 5 p1[[2]] + 4 p2[[2]] - p3[[2]]) +      t^3 (-p0[[2]] + 3 p1[[2]] - 3 p2[[2]] + p3[[2]]),     2 p1[[3]] + t (p2[[3]] - p0[[3]]) +      t^2 (2 p0[[3]] - 5 p1[[3]] + 4 p2[[3]] - p3[[3]]) +      t^3 (-p0[[3]] + 3 p1[[3]] - 3 p2[[3]] + p3[[3]])};  offset = 0;step = 0.25crOneOffsets = {0 + offset, -Pi/16 + offset, -Pi/28 + offset,    Pi/8 + offset};crOneTs =   Range[0, 2 Pi - step,    step];                            (*curve31 上的点*)crOneConnectors = Table[random = RandomReal[{1, 15}]*0.02;   p1 = {2 Cos[t + crOneOffsets[[1]] + random],      1.5 Sin[t + crOneOffsets[[1]] + random],      0.2 Sin[2 (t + crOneOffsets[[1]]) + 3 Pi/2] + 3};   p2 = {4 Cos[t + crOneOffsets[[2]]], 4.2 Sin[t + crOneOffsets[[2]]],      0.6 Sin[2 (t + crOneOffsets[[2]]) + 3 Pi/2] + 3.2};   p3 = {4.2 Cos[t + crOneOffsets[[3]]],      4.4 Sin[t + crOneOffsets[[3]]],      0.6 Sin[2 (t + crOneOffsets[[3]]) + 3 Pi/2] + 2.5};   p4 = {3.4 Cos[t + crOneOffsets[[4]] + random],      3.8 Sin[t + crOneOffsets[[4]] + random], 0};   pts = {p1, p2, p3, p4};   newPts =     Join[{2.1 pts[[1]] - pts[[2]]},      pts, {2.2 pts[[-1]] - pts[[-2]]}];   crPts =     Flatten[Table[      Table[catmullRom3D[newPts[[i ;; i + 3]], s], {s, 0, 1,         1/30}], {i, 1, Length[newPts] - 3}], 1];   (*4. 绘制这一段的 CR 曲线+原控制点*){Thick, Opacity[0.7], Blue, Line[crPts],     Black, PointSize[0.015], Point[pts]}, {t, crOneTs}];  png = Show[Graphics3D[crOneConnectors], curve12, curve21, curve22,   curve31, axes, Axes -> True, AxesLabel -> {"X", "Y", "Z"},   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1},   ImageSize -> Large, PlotLabel -> "Bird's Nest"]
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mathematica 以此类推，绘制更多
Code
curve12 =   ParametricPlot3D[{2 Cos[t], 1.5 Sin[t],     0.2 Sin[2 t + Pi + Pi/2] + 3}, {t, 0, 2 Pi},    PlotStyle -> {Thick, Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];curve21 =   ParametricPlot3D[{4 Cos[t], 4.2 Sin[t],     0.6 Sin[2 t + Pi + Pi/2] + 3.2}, {t, 0, 2 Pi},    PlotStyle -> {Thick, Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];curve22 =   ParametricPlot3D[{4.2 Cos[t], 4.4 Sin[t],     0.6 Sin[2 t + Pi + Pi/2] + 2.5}, {t, 0, 2 Pi},    PlotStyle -> {Thick, Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];curve31 =   ParametricPlot3D[{3.4 Cos[t], 3.8 Sin[t], 0}, {t, 0, 2 \[Pi]},    PlotStyle -> {Thick, Blue}, Axes -> False,    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}]; axes = Graphics3D[{Arrowheads[0.05], {Red,      Arrow[{{0, 0, 0}, {5, 0, 0}}]},(*X 轴*){Green,      Arrow[{{0, 0, 0}, {0, 5, 0}}]},(*Y 轴*){Blue,      Arrow[{{0, 0, 0}, {0, 0, 5}}]}    (*Z 轴*)}]; catmullRom3D[{p0_, p1_, p2_, p3_}, t_] :=   0.5*{2 p1[[1]] + t (p2[[1]] - p0[[1]]) +      t^2 (2 p0[[1]] - 5 p1[[1]] + 4 p2[[1]] - p3[[1]]) +      t^3 (-p0[[1]] + 3 p1[[1]] - 3 p2[[1]] + p3[[1]]),     2 p1[[2]] + t (p2[[2]] - p0[[2]]) +      t^2 (2 p0[[2]] - 5 p1[[2]] + 4 p2[[2]] - p3[[2]]) +      t^3 (-p0[[2]] + 3 p1[[2]] - 3 p2[[2]] + p3[[2]]),     2 p1[[3]] + t (p2[[3]] - p0[[3]]) +      t^2 (2 p0[[3]] - 5 p1[[3]] + 4 p2[[3]] - p3[[3]]) +      t^3 (-p0[[3]] + 3 p1[[3]] - 3 p2[[3]] + p3[[3]])};  offset = 0;step = 0.25crOneOffsets = {0 + offset, -Pi/16 + offset, -Pi/28 + offset,    Pi/8 + offset};crOneTs =   Range[0, 2 Pi - step,    step];                            (*curve31 上的点*)crOneConnectors = Table[random = RandomReal[{1, 15}]*0.02;   p1 = {2 Cos[t + crOneOffsets[[1]] + random],      1.5 Sin[t + crOneOffsets[[1]] + random],      0.2 Sin[2 (t + crOneOffsets[[1]]) + 3 Pi/2] + 3};   p2 = {4 Cos[t + crOneOffsets[[2]]], 4.2 Sin[t + crOneOffsets[[2]]],      0.6 Sin[2 (t + crOneOffsets[[2]]) + 3 Pi/2] + 3.2};   p3 = {4.2 Cos[t + crOneOffsets[[3]]],      4.4 Sin[t + crOneOffsets[[3]]],      0.6 Sin[2 (t + crOneOffsets[[3]]) + 3 Pi/2] + 2.5};   p4 = {3.4 Cos[t + crOneOffsets[[4]] + random],      3.8 Sin[t + crOneOffsets[[4]] + random], 0};   pts = {p1, p2, p3, p4};   newPts =     Join[{2.1 pts[[1]] - pts[[2]]},      pts, {2.2 pts[[-1]] - pts[[-2]]}];   crPts =     Flatten[Table[      Table[catmullRom3D[newPts[[i ;; i + 3]], s], {s, 0, 1,         1/30}], {i, 1, Length[newPts] - 3}], 1];   (*4. 绘制这一段的 CR 曲线+原控制点*){Thick, Opacity[0.7], Blue, Line[crPts],     Black, PointSize[0.015], Point[pts]}, {t, crOneTs}]; crOneConnectors1 = Table[random = RandomReal[{1, 15}]*0.02;   p1 = {2 Cos[t + crOneOffsets[[1]] + random],      1.5 Sin[t + crOneOffsets[[1]] + random],      0.2 Sin[2 (t + crOneOffsets[[1]] + random) + 3 Pi/2] + 3};   p2 = {4 Cos[t + crOneOffsets[[2]]], 4.2 Sin[t + crOneOffsets[[2]]],      0.6 Sin[2 (t + crOneOffsets[[2]]) + 3 Pi/2] + 3.2};   p3 = {4.2 Cos[t + crOneOffsets[[3]] + step],      4.4 Sin[t + crOneOffsets[[3]] + step],      0.6 Sin[2 (t + crOneOffsets[[3]] + step) + 3 Pi/2] + 2.5};   p4 = {3.4 Cos[t + crOneOffsets[[4]] + random + step],      3.8 Sin[t + crOneOffsets[[4]] + random + step], 0};   pts = {p1, p2, p3, p4};   crPts =     Flatten[Table[      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i,        1, Length[pts] - 3}], 1];   (*4. 绘制这一段的 CR 曲线+原控制点*){Thick, Opacity[0.7], Blue, Line[crPts],     Transparent, PointSize[0.001], Point[pts]}, {t, crOneTs}]; crOneConnectors2 = Table[random = RandomReal[{1, 15}]*0.02;   p1 = {2 Cos[t + crOneOffsets[[1]] + random],      1.5 Sin[t + crOneOffsets[[1]] + random],      0.2 Sin[2 (t + crOneOffsets[[1]] + random) + 3 Pi/2] + 3};   p2 = {4 Cos[t + crOneOffsets[[2]] + step],      4.2 Sin[t + crOneOffsets[[2]] + step],      0.6 Sin[2 (t + crOneOffsets[[2]] + step) + 3 Pi/2] + 3.2};   p3 = {4.2 Cos[t + crOneOffsets[[3]]],      4.4 Sin[t + crOneOffsets[[3]]],      0.6 Sin[2 (t + crOneOffsets[[3]]) + 3 Pi/2] + 2.5};   p4 = {3.4 Cos[t + crOneOffsets[[4]] + random - step],      3.8 Sin[t + crOneOffsets[[4]] + random - step], 0};   pts = {p1, p2, p3, p4};   crPts =     Flatten[Table[      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i,        1, Length[pts] - 3}], 1];   (*4. 绘制这一段的 CR 曲线+原控制点*){Thick, Opacity[0.7], Blue, Line[crPts],     Transparent, PointSize[0.001], Point[pts]}, {t, crOneTs}];  offset = Pi/32;crTwoOffsets = {0 + offset, Pi/16 + offset,    Pi/28 + offset, -Pi/8 + offset};crTwoTs = Range[0, 2 Pi, step];crTwoConnectors = Table[random = RandomReal[{1, 15}]*0.02;   p1 = {2 Cos[t + crTwoOffsets[[1]] + random],      1.5 Sin[t + crTwoOffsets[[1]] + random],      0.2 Sin[2 (t + crTwoOffsets[[1]]) + 3 Pi/2] + 3};   p2 = {4 Cos[t + crTwoOffsets[[2]]], 4.2 Sin[t + crTwoOffsets[[2]]],      0.6 Sin[2 (t + crTwoOffsets[[2]]) + 3 Pi/2] + 3.2};   p3 = {4.2 Cos[t + crTwoOffsets[[3]]],      4.4 Sin[t + crTwoOffsets[[3]]],      0.6 Sin[2 (t + crTwoOffsets[[3]]) + 3 Pi/2] + 2.5};   p4 = {3.4 Cos[t + crTwoOffsets[[4]] + random],      3.8 Sin[t + crTwoOffsets[[4]] + random], 0};   pts = {p1, p2, p3, p4};   newPts =     Join[{2.1 pts[[1]] - pts[[2]]},(*p0=2*p1-p2*)     pts, {2.2 pts[[-1]] - pts[[-2]]}];   crPts =     Flatten[Table[      Table[catmullRom3D[newPts[[i ;; i + 3]], s], {s, 0, 1,         1/30}], {i, 1, Length[newPts] - 3}], 1];   {Thick, Opacity[0.7], Blue, Line[crPts], Transparent,     PointSize[0.001], Point[pts]}, {t, crTwoTs}]; crTwoConnectors1 = Table[random = RandomReal[{1, 15}]*0.02;   p1 = {2 Cos[t + crTwoOffsets[[1]] + random],      1.5 Sin[t + crTwoOffsets[[1]] + random],      0.2 Sin[2 (t + crTwoOffsets[[1]]) + 3 Pi/2] + 3};   p2 = {4 Cos[t + crTwoOffsets[[2]]], 4.2 Sin[t + crTwoOffsets[[2]]],      0.6 Sin[2 (t + crTwoOffsets[[2]]) + 3 Pi/2] + 3.2};   p3 = {4.2 Cos[t + crTwoOffsets[[3]] + step],      4.4 Sin[t + crTwoOffsets[[3]] + step],      0.6 Sin[2 (t + crTwoOffsets[[3]] + step) + 3 Pi/2] + 2.5};   p4 = {3.4 Cos[t + crTwoOffsets[[4]] + random + step],      3.8 Sin[t + crTwoOffsets[[4]] + random + step], 0};   pts = {p1, p2, p3, p4};   crPts =     Flatten[Table[      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i,        1, Length[pts] - 3}], 1];   {Thick, Opacity[0.7], Blue, Line[crPts], Transparent,     PointSize[0.001], Point[pts]}, {t, crTwoTs}];  crTwoConnectors2 = Table[random = RandomReal[{1, 15}]*0.02;   p1 = {2 Cos[t + crTwoOffsets[[1]] + random],      1.5 Sin[t + crTwoOffsets[[1]] + random],      0.2 Sin[2 (t + crTwoOffsets[[1]]) + 3 Pi/2] + 3};   p2 = {4 Cos[t + crTwoOffsets[[2]] + step],      4.2 Sin[t + crTwoOffsets[[2]] + step],      0.6 Sin[2 (t + crTwoOffsets[[2]] + step) + 3 Pi/2] + 3.2};   p3 = {4.2 Cos[t + crTwoOffsets[[3]]],      4.4 Sin[t + crTwoOffsets[[3]]],      0.6 Sin[2 (t + crTwoOffsets[[3]]) + 3 Pi/2] + 2.5};   p4 = {3.4 Cos[t + crTwoOffsets[[4]] + random - step],      3.8 Sin[t + crTwoOffsets[[4]] + random - step], 0};   pts = {p1, p2, p3, p4};   crPts =     Flatten[Table[      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i,        1, Length[pts] - 3}], 1];   {Thick, Opacity[0.7], Blue, Line[crPts], Transparent,     PointSize[0.001], Point[pts]}, {t, crTwoTs}];      png = Show[Graphics3D[crOneConnectors], Graphics3D[crOneConnectors1],   Graphics3D[crOneConnectors2], Graphics3D[crTwoConnectors],   Graphics3D[crTwoConnectors1], Graphics3D[crTwoConnectors2], curve12,   curve21, curve22, curve31, axes, Axes -> True,   AxesLabel -> {"X", "Y", "Z"},   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1},   ImageSize -> Large, PlotLabel -> "Bird's Nest"]
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mathematica 调整下图形
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curve12 =   ParametricPlot3D[{2 Cos[t], 1.5 Sin[t],     0.2 Sin[2 t + Pi + Pi/2] + 3}, {t, 0, 2 Pi},    PlotStyle -> {Thickness[0.002], Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];curve21 =   ParametricPlot3D[{4 Cos[t], 4.2 Sin[t],     0.6 Sin[2 t + Pi + Pi/2] + 3.2}, {t, 0, 2 Pi},    PlotStyle -> {Thickness[0.002], Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];curve22 =   ParametricPlot3D[{4.2 Cos[t], 4.4 Sin[t],     0.6 Sin[2 t + Pi + Pi/2] + 2.5}, {t, 0, 2 Pi},    PlotStyle -> {Thickness[0.002], Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];curve31 =   ParametricPlot3D[{3.4 Cos[t], 3.8 Sin[t], 0}, {t, 0, 2 \[Pi]},    PlotStyle -> {Thickness[0.002], Blue}, Axes -> False,    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}]; axes = Graphics3D[{Arrowheads[0.05], {Red,      Arrow[{{0, 0, 0}, {5, 0, 0}}]},(*X 轴*){Green,      Arrow[{{0, 0, 0}, {0, 5, 0}}]},(*Y 轴*){Blue,      Arrow[{{0, 0, 0}, {0, 0, 5}}]}    (*Z 轴*)}]; catmullRom3D[{p0_, p1_, p2_, p3_}, t_] :=   0.5*{2 p1[[1]] + t (p2[[1]] - p0[[1]]) +      t^2 (2 p0[[1]] - 5 p1[[1]] + 4 p2[[1]] - p3[[1]]) +      t^3 (-p0[[1]] + 3 p1[[1]] - 3 p2[[1]] + p3[[1]]),     2 p1[[2]] + t (p2[[2]] - p0[[2]]) +      t^2 (2 p0[[2]] - 5 p1[[2]] + 4 p2[[2]] - p3[[2]]) +      t^3 (-p0[[2]] + 3 p1[[2]] - 3 p2[[2]] + p3[[2]]),     2 p1[[3]] + t (p2[[3]] - p0[[3]]) +      t^2 (2 p0[[3]] - 5 p1[[3]] + 4 p2[[3]] - p3[[3]]) +      t^3 (-p0[[3]] + 3 p1[[3]] - 3 p2[[3]] + p3[[3]])};  offset = 0;step = 0.25crOneOffsets = {0 + offset, -Pi/16 + offset, -Pi/28 + offset,    Pi/8 + offset};crOneTs =   Range[0, 2 Pi - step,    step];                            (*curve31 上的点*)crOneConnectors = Table[random = RandomReal[{1, 15}]*0.02;   p1 = {2 Cos[t + crOneOffsets[[1]] + random],      1.5 Sin[t + crOneOffsets[[1]] + random],      0.2 Sin[2 (t + crOneOffsets[[1]]) + 3 Pi/2] + 3};   p2 = {4 Cos[t + crOneOffsets[[2]]], 4.2 Sin[t + crOneOffsets[[2]]],      0.6 Sin[2 (t + crOneOffsets[[2]]) + 3 Pi/2] + 3.2};   p3 = {4.2 Cos[t + crOneOffsets[[3]]],      4.4 Sin[t + crOneOffsets[[3]]],      0.6 Sin[2 (t + crOneOffsets[[3]]) + 3 Pi/2] + 2.5};   p4 = {3.4 Cos[t + crOneOffsets[[4]] + random],      3.8 Sin[t + crOneOffsets[[4]] + random], 0};   pts = {p1, p2, p3, p4};   newPts =     Join[{2.1 pts[[1]] - pts[[2]]},      pts, {2.2 pts[[-1]] - pts[[-2]]}];   crPts =     Flatten[Table[      Table[catmullRom3D[newPts[[i ;; i + 3]], s], {s, 0, 1,         1/30}], {i, 1, Length[newPts] - 3}], 1];   (*4. 绘制这一段的 CR 曲线+原控制点*){Thickness[0.002], Opacity[0.7], Blue,     Line[crPts], Transparent, PointSize[0.015], Point[pts]}, {t,     crOneTs}]; crOneConnectors1 = Table[random = RandomReal[{1, 15}]*0.02;   p1 = {2 Cos[t + crOneOffsets[[1]] + random],      1.5 Sin[t + crOneOffsets[[1]] + random],      0.2 Sin[2 (t + crOneOffsets[[1]] + random) + 3 Pi/2] + 3};   p2 = {4 Cos[t + crOneOffsets[[2]]], 4.2 Sin[t + crOneOffsets[[2]]],      0.6 Sin[2 (t + crOneOffsets[[2]]) + 3 Pi/2] + 3.2};   p3 = {4.2 Cos[t + crOneOffsets[[3]] + step],      4.4 Sin[t + crOneOffsets[[3]] + step],      0.6 Sin[2 (t + crOneOffsets[[3]] + step) + 3 Pi/2] + 2.5};   p4 = {3.4 Cos[t + crOneOffsets[[4]] + random + step],      3.8 Sin[t + crOneOffsets[[4]] + random + step], 0};   pts = {p1, p2, p3, p4};   crPts =     Flatten[Table[      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i,        1, Length[pts] - 3}], 1];   (*4. 绘制这一段的 CR 曲线+原控制点*){Thickness[0.002], Opacity[0.7], Blue,     Line[crPts], Transparent, PointSize[0.001], Point[pts]}, {t,     crOneTs}]; crOneConnectors2 = Table[random = RandomReal[{1, 15}]*0.02;   p1 = {2 Cos[t + crOneOffsets[[1]] + random],      1.5 Sin[t + crOneOffsets[[1]] + random],      0.2 Sin[2 (t + crOneOffsets[[1]] + random) + 3 Pi/2] + 3};   p2 = {4 Cos[t + crOneOffsets[[2]] + step],      4.2 Sin[t + crOneOffsets[[2]] + step],      0.6 Sin[2 (t + crOneOffsets[[2]] + step) + 3 Pi/2] + 3.2};   p3 = {4.2 Cos[t + crOneOffsets[[3]]],      4.4 Sin[t + crOneOffsets[[3]]],      0.6 Sin[2 (t + crOneOffsets[[3]]) + 3 Pi/2] + 2.5};   p4 = {3.4 Cos[t + crOneOffsets[[4]] + random - step],      3.8 Sin[t + crOneOffsets[[4]] + random - step], 0};   pts = {p1, p2, p3, p4};   crPts =     Flatten[Table[      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i,        1, Length[pts] - 3}], 1];   (*4. 绘制这一段的 CR 曲线+原控制点*){Thickness[0.002], Opacity[0.7], Blue,     Line[crPts], Transparent, PointSize[0.001], Point[pts]}, {t,     crOneTs}];  offset = Pi/32;crTwoOffsets = {0 + offset, Pi/16 + offset,    Pi/28 + offset, -Pi/8 + offset};crTwoTs = Range[0, 2 Pi, step];crTwoConnectors = Table[random = RandomReal[{1, 15}]*0.02;   p1 = {2 Cos[t + crTwoOffsets[[1]] + random],      1.5 Sin[t + crTwoOffsets[[1]] + random],      0.2 Sin[2 (t + crTwoOffsets[[1]]) + 3 Pi/2] + 3};   p2 = {4 Cos[t + crTwoOffsets[[2]]], 4.2 Sin[t + crTwoOffsets[[2]]],      0.6 Sin[2 (t + crTwoOffsets[[2]]) + 3 Pi/2] + 3.2};   p3 = {4.2 Cos[t + crTwoOffsets[[3]]],      4.4 Sin[t + crTwoOffsets[[3]]],      0.6 Sin[2 (t + crTwoOffsets[[3]]) + 3 Pi/2] + 2.5};   p4 = {3.4 Cos[t + crTwoOffsets[[4]] + random],      3.8 Sin[t + crTwoOffsets[[4]] + random], 0};   pts = {p1, p2, p3, p4};   newPts =     Join[{2.1 pts[[1]] - pts[[2]]},(*p0=2*p1-p2*)     pts, {2.2 pts[[-1]] - pts[[-2]]}];   crPts =     Flatten[Table[      Table[catmullRom3D[newPts[[i ;; i + 3]], s], {s, 0, 1,         1/30}], {i, 1, Length[newPts] - 3}], 1];   {Thickness[0.002], Opacity[0.7], Blue, Line[crPts], Transparent,     PointSize[0.001], Point[pts]}, {t, crTwoTs}]; crTwoConnectors1 = Table[random = RandomReal[{1, 15}]*0.02;   p1 = {2 Cos[t + crTwoOffsets[[1]] + random],      1.5 Sin[t + crTwoOffsets[[1]] + random],      0.2 Sin[2 (t + crTwoOffsets[[1]]) + 3 Pi/2] + 3};   p2 = {4 Cos[t + crTwoOffsets[[2]]], 4.2 Sin[t + crTwoOffsets[[2]]],      0.6 Sin[2 (t + crTwoOffsets[[2]]) + 3 Pi/2] + 3.2};   p3 = {4.2 Cos[t + crTwoOffsets[[3]] + step],      4.4 Sin[t + crTwoOffsets[[3]] + step],      0.6 Sin[2 (t + crTwoOffsets[[3]] + step) + 3 Pi/2] + 2.5};   p4 = {3.4 Cos[t + crTwoOffsets[[4]] + random + step],      3.8 Sin[t + crTwoOffsets[[4]] + random + step], 0};   pts = {p1, p2, p3, p4};   crPts =     Flatten[Table[      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i,        1, Length[pts] - 3}], 1];   {Thickness[0.002], Opacity[0.7], Blue, Line[crPts], Transparent,     PointSize[0.001], Point[pts]}, {t, crTwoTs}];  crTwoConnectors2 = Table[random = RandomReal[{1, 15}]*0.02;   p1 = {2 Cos[t + crTwoOffsets[[1]] + random],      1.5 Sin[t + crTwoOffsets[[1]] + random],      0.2 Sin[2 (t + crTwoOffsets[[1]]) + 3 Pi/2] + 3};   p2 = {4 Cos[t + crTwoOffsets[[2]] + step],      4.2 Sin[t + crTwoOffsets[[2]] + step],      0.6 Sin[2 (t + crTwoOffsets[[2]] + step) + 3 Pi/2] + 3.2};   p3 = {4.2 Cos[t + crTwoOffsets[[3]]],      4.4 Sin[t + crTwoOffsets[[3]]],      0.6 Sin[2 (t + crTwoOffsets[[3]]) + 3 Pi/2] + 2.5};   p4 = {3.4 Cos[t + crTwoOffsets[[4]] + random - step],      3.8 Sin[t + crTwoOffsets[[4]] + random - step], 0};   pts = {p1, p2, p3, p4};   crPts =     Flatten[Table[      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i,        1, Length[pts] - 3}], 1];   {Thickness[0.002], Opacity[0.7], Blue, Line[crPts], Transparent,     PointSize[0.001], Point[pts]}, {t, crTwoTs}];    png = Show[Graphics3D[crOneConnectors], Graphics3D[crOneConnectors1],   Graphics3D[crOneConnectors2], Graphics3D[crTwoConnectors],   Graphics3D[crTwoConnectors1], Graphics3D[crTwoConnectors2],   curve12,(*curve21,curve22,*)curve31, axes, Axes -> True,   AxesLabel -> {"X", "Y", "Z"},   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1},   ImageSize -> Large, PlotLabel -> "Bird's Nest"]
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mathematica 随机添加一些横线
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curve12 =   ParametricPlot3D[{2 Cos[t], 1.5 Sin[t],     0.2 Sin[2 t + Pi + Pi/2] + 3}, {t, 0, 2 Pi},    PlotStyle -> {Thickness[0.002], Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];curve21 =   ParametricPlot3D[{4 Cos[t], 4.2 Sin[t],     0.6 Sin[2 t + Pi + Pi/2] + 3.2}, {t, 0, 2 Pi},    PlotStyle -> {Thickness[0.002], Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];curve22 =   ParametricPlot3D[{4.2 Cos[t], 4.4 Sin[t],     0.6 Sin[2 t + Pi + Pi/2] + 2.5}, {t, 0, 2 Pi},    PlotStyle -> {Thickness[0.002], Blue},    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];curve31 =   ParametricPlot3D[{3.4 Cos[t], 3.8 Sin[t], 0}, {t, 0, 2 \[Pi]},    PlotStyle -> {Thickness[0.002], Blue}, Axes -> False,    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}]; axes = Graphics3D[{Arrowheads[0.05], {Red,      Arrow[{{0, 0, 0}, {5, 0, 0}}]},(*X 轴*){Green,      Arrow[{{0, 0, 0}, {0, 5, 0}}]},(*Y 轴*){Blue,      Arrow[{{0, 0, 0}, {0, 0, 5}}]}    (*Z 轴*)}]; catmullRom3D[{p0_, p1_, p2_, p3_}, t_] :=   0.5*{2 p1[[1]] + t (p2[[1]] - p0[[1]]) +      t^2 (2 p0[[1]] - 5 p1[[1]] + 4 p2[[1]] - p3[[1]]) +      t^3 (-p0[[1]] + 3 p1[[1]] - 3 p2[[1]] + p3[[1]]),     2 p1[[2]] + t (p2[[2]] - p0[[2]]) +      t^2 (2 p0[[2]] - 5 p1[[2]] + 4 p2[[2]] - p3[[2]]) +      t^3 (-p0[[2]] + 3 p1[[2]] - 3 p2[[2]] + p3[[2]]),     2 p1[[3]] + t (p2[[3]] - p0[[3]]) +      t^2 (2 p0[[3]] - 5 p1[[3]] + 4 p2[[3]] - p3[[3]]) +      t^3 (-p0[[3]] + 3 p1[[3]] - 3 p2[[3]] + p3[[3]])}; p1Collection = {};p2Collection = {};p3Collection = {};p4Collection = {}; offset = 0;step = 0.25crOneOffsets = {0 + offset, -Pi/16 + offset, -Pi/28 + offset,    Pi/8 + offset};crOneTs =   Range[0, 2 Pi - step,    step];                            (*curve31 上的点*)crOneConnectors = Table[random = RandomReal[{1, 15}]*0.02;   p1 = {2 Cos[t + crOneOffsets[[1]] + random],      1.5 Sin[t + crOneOffsets[[1]] + random],      0.2 Sin[2 (t + crOneOffsets[[1]]) + 3 Pi/2] + 3};   p2 = {4 Cos[t + crOneOffsets[[2]]], 4.2 Sin[t + crOneOffsets[[2]]],      0.6 Sin[2 (t + crOneOffsets[[2]]) + 3 Pi/2] + 3.2};   p3 = {4.2 Cos[t + crOneOffsets[[3]]],      4.4 Sin[t + crOneOffsets[[3]]],      0.6 Sin[2 (t + crOneOffsets[[3]]) + 3 Pi/2] + 2.5};   p4 = {3.4 Cos[t + crOneOffsets[[4]] + random],      3.8 Sin[t + crOneOffsets[[4]] + random], 0};   AppendTo[p1Collection, p1];   AppendTo[p2Collection, p2];   AppendTo[p3Collection, p3];   AppendTo[p4Collection, p4];   pts = {p1, p2, p3, p4};   newPts =     Join[{2.1 pts[[1]] - pts[[2]]},      pts, {2.2 pts[[-1]] - pts[[-2]]}];   crPts =     Flatten[Table[      Table[catmullRom3D[newPts[[i ;; i + 3]], s], {s, 0, 1,         1/30}], {i, 1, Length[newPts] - 3}], 1];   (*4. 绘制这一段的 CR 曲线+原控制点*){Thickness[0.002], Opacity[0.7], Blue,     Line[crPts], Transparent, PointSize[0.015], Point[pts]}, {t,     crOneTs}]; crOneConnectors1 = Table[random = RandomReal[{1, 15}]*0.02;   p1 = {2 Cos[t + crOneOffsets[[1]] + random],      1.5 Sin[t + crOneOffsets[[1]] + random],      0.2 Sin[2 (t + crOneOffsets[[1]] + random) + 3 Pi/2] + 3};   p2 = {4 Cos[t + crOneOffsets[[2]]], 4.2 Sin[t + crOneOffsets[[2]]],      0.6 Sin[2 (t + crOneOffsets[[2]]) + 3 Pi/2] + 3.2};   p3 = {4.2 Cos[t + crOneOffsets[[3]] + step],      4.4 Sin[t + crOneOffsets[[3]] + step],      0.6 Sin[2 (t + crOneOffsets[[3]] + step) + 3 Pi/2] + 2.5};   p4 = {3.4 Cos[t + crOneOffsets[[4]] + random + step],      3.8 Sin[t + crOneOffsets[[4]] + random + step], 0};   pts = {p1, p2, p3, p4};   crPts =     Flatten[Table[      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i,        1, Length[pts] - 3}], 1];   (*4. 绘制这一段的 CR 曲线+原控制点*){Thickness[0.002], Opacity[0.7], Blue,     Line[crPts], Transparent, PointSize[0.001], Point[pts]}, {t,     crOneTs}]; crOneConnectors2 = Table[random = RandomReal[{1, 15}]*0.02;   p1 = {2 Cos[t + crOneOffsets[[1]] + random],      1.5 Sin[t + crOneOffsets[[1]] + random],      0.2 Sin[2 (t + crOneOffsets[[1]] + random) + 3 Pi/2] + 3};   p2 = {4 Cos[t + crOneOffsets[[2]] + step],      4.2 Sin[t + crOneOffsets[[2]] + step],      0.6 Sin[2 (t + crOneOffsets[[2]] + step) + 3 Pi/2] + 3.2};   p3 = {4.2 Cos[t + crOneOffsets[[3]]],      4.4 Sin[t + crOneOffsets[[3]]],      0.6 Sin[2 (t + crOneOffsets[[3]]) + 3 Pi/2] + 2.5};   p4 = {3.4 Cos[t + crOneOffsets[[4]] + random - step],      3.8 Sin[t + crOneOffsets[[4]] + random - step], 0};   pts = {p1, p2, p3, p4};   crPts =     Flatten[Table[      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i,        1, Length[pts] - 3}], 1];   (*4. 绘制这一段的 CR 曲线+原控制点*){Thickness[0.002], Opacity[0.7], Blue,     Line[crPts], Transparent, PointSize[0.001], Point[pts]}, {t,     crOneTs}]; (*生成30条随机连接线，确保位置差不超过30*)crOneRandomP1P2Lines =   Table[(*随机选择p1集合中的索引*)   idx1 = RandomInteger[{1, Length[p1Collection]}];   (*计算p2集合中的有效索引范围*)minIdx2 = Max[1, idx1 - 1];   maxIdx2 = Min[Length[p2Collection], idx1 + 6];   (*在有效范围内随机选择p2索引*)idx2 = RandomInteger[{minIdx2, maxIdx2}];   (*获取点*)randomP1 = p1Collection[[idx1]];   randomP2 = p2Collection[[idx2]];   (*计算位置差*)posDiff = Abs[idx1 - idx2];   (*创建颜色-位置差越小越蓝，越大越红*)hueValue = Rescale[posDiff, {0, 10}, {0.6, 0}];   (*lineColor=Hue[hueValue];*)lineColor = Blue;   (*绘制线段和端点*){Thickness[0.002], Opacity[0.7], lineColor,     Line[{randomP1, randomP2}]}, {30}]; (*循环30次*) crOneRandomP2Lines =   Table[(*随机选择p1集合中的索引*)   idx1 = RandomInteger[{1, Length[p2Collection]}];   (*计算p2集合中的有效索引范围*)minIdx2 = Max[1, idx1 - 1];   maxIdx2 = Min[Length[p2Collection], idx1 + 6];   (*在有效范围内随机选择p2索引*)idx2 = RandomInteger[{minIdx2, maxIdx2}];   (*获取点*)randomP1 = p2Collection[[idx1]];   randomP2 = p2Collection[[idx2]];   (*计算位置差*)posDiff = Abs[idx1 - idx2];   (*创建颜色-位置差越小越蓝，越大越红*)hueValue = Rescale[posDiff, {0, 10}, {0.6, 0}];   lineColor = Blue;   (*绘制线段和端点*){Thickness[0.002], Opacity[0.7], lineColor,     Line[{randomP1, randomP2}]}, {30}]; (*循环30次*) crOneRandomP3P4Lines =   Table[(*随机选择p1集合中的索引*)   idx1 = RandomInteger[{1, Length[p3Collection]}];   (*计算p2集合中的有效索引范围*)minIdx2 = Max[1, idx1 - 8];   maxIdx2 = Min[Length[p4Collection], idx1];   (*在有效范围内随机选择p2索引*)idx2 = RandomInteger[{minIdx2, maxIdx2}];   (*获取点*)randomP1 = p3Collection[[idx1]];   randomP2 = p4Collection[[idx2]];   (*计算位置差*)posDiff = Abs[idx1 - idx2];   (*创建颜色-位置差越小越蓝，越大越红*)hueValue = Rescale[posDiff, {0, 10}, {0.6, 0}];   lineColor = Blue;   (*绘制线段和端点*){Thickness[0.002], Opacity[0.7], lineColor,     Line[{randomP1, randomP2}]}, {30}]; (*循环30次*) p1Collection = {};p2Collection = {};p3Collection = {};p4Collection = {};offset = Pi/32;crTwoOffsets = {0 + offset, Pi/16 + offset,    Pi/28 + offset, -Pi/8 + offset};crTwoTs = Range[0, 2 Pi, step];crTwoConnectors = Table[random = RandomReal[{1, 15}]*0.02;   p1 = {2 Cos[t + crTwoOffsets[[1]] + random],      1.5 Sin[t + crTwoOffsets[[1]] + random],      0.2 Sin[2 (t + crTwoOffsets[[1]]) + 3 Pi/2] + 3};   p2 = {4 Cos[t + crTwoOffsets[[2]]], 4.2 Sin[t + crTwoOffsets[[2]]],      0.6 Sin[2 (t + crTwoOffsets[[2]]) + 3 Pi/2] + 3.2};   p3 = {4.2 Cos[t + crTwoOffsets[[3]]],      4.4 Sin[t + crTwoOffsets[[3]]],      0.6 Sin[2 (t + crTwoOffsets[[3]]) + 3 Pi/2] + 2.5};   p4 = {3.4 Cos[t + crTwoOffsets[[4]] + random],      3.8 Sin[t + crTwoOffsets[[4]] + random], 0};   AppendTo[p1Collection, p1];   AppendTo[p2Collection, p2];   AppendTo[p3Collection, p3];   AppendTo[p4Collection, p4];   pts = {p1, p2, p3, p4};   newPts =     Join[{2.1 pts[[1]] - pts[[2]]},(*p0=2*p1-p2*)     pts, {2.2 pts[[-1]] - pts[[-2]]}];   crPts =     Flatten[Table[      Table[catmullRom3D[newPts[[i ;; i + 3]], s], {s, 0, 1,         1/30}], {i, 1, Length[newPts] - 3}], 1];   {Thickness[0.002], Opacity[0.7], Blue, Line[crPts], Transparent,     PointSize[0.001], Point[pts]}, {t, crTwoTs}]; crTwoConnectors1 = Table[random = RandomReal[{1, 15}]*0.02;   p1 = {2 Cos[t + crTwoOffsets[[1]] + random],      1.5 Sin[t + crTwoOffsets[[1]] + random],      0.2 Sin[2 (t + crTwoOffsets[[1]]) + 3 Pi/2] + 3};   p2 = {4 Cos[t + crTwoOffsets[[2]]], 4.2 Sin[t + crTwoOffsets[[2]]],      0.6 Sin[2 (t + crTwoOffsets[[2]]) + 3 Pi/2] + 3.2};   p3 = {4.2 Cos[t + crTwoOffsets[[3]] + step],      4.4 Sin[t + crTwoOffsets[[3]] + step],      0.6 Sin[2 (t + crTwoOffsets[[3]] + step) + 3 Pi/2] + 2.5};   p4 = {3.4 Cos[t + crTwoOffsets[[4]] + random + step],      3.8 Sin[t + crTwoOffsets[[4]] + random + step], 0};   pts = {p1, p2, p3, p4};   crPts =     Flatten[Table[      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i,        1, Length[pts] - 3}], 1];   {Thickness[0.002], Opacity[0.7], Blue, Line[crPts], Transparent,     PointSize[0.001], Point[pts]}, {t, crTwoTs}];  crTwoConnectors2 = Table[random = RandomReal[{1, 15}]*0.02;   p1 = {2 Cos[t + crTwoOffsets[[1]] + random],      1.5 Sin[t + crTwoOffsets[[1]] + random],      0.2 Sin[2 (t + crTwoOffsets[[1]]) + 3 Pi/2] + 3};   p2 = {4 Cos[t + crTwoOffsets[[2]] + step],      4.2 Sin[t + crTwoOffsets[[2]] + step],      0.6 Sin[2 (t + crTwoOffsets[[2]] + step) + 3 Pi/2] + 3.2};   p3 = {4.2 Cos[t + crTwoOffsets[[3]]],      4.4 Sin[t + crTwoOffsets[[3]]],      0.6 Sin[2 (t + crTwoOffsets[[3]]) + 3 Pi/2] + 2.5};   p4 = {3.4 Cos[t + crTwoOffsets[[4]] + random - step],      3.8 Sin[t + crTwoOffsets[[4]] + random - step], 0};   pts = {p1, p2, p3, p4};   crPts =     Flatten[Table[      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i,        1, Length[pts] - 3}], 1];   {Thickness[0.002], Opacity[0.7], Blue, Line[crPts], Transparent,     PointSize[0.001], Point[pts]}, {t, crTwoTs}]; (*生成30条随机连接线，确保位置差不超过30*)crTwoRandomP1P2Lines =   Table[(*随机选择p1集合中的索引*)   idx1 = RandomInteger[{1, Length[p1Collection]}];   (*计算p2集合中的有效索引范围*)minIdx2 = Max[1, idx1 - 6];   maxIdx2 = Min[Length[p2Collection], idx1 + 1];   (*在有效范围内随机选择p2索引*)idx2 = RandomInteger[{minIdx2, maxIdx2}];   (*获取点*)randomP1 = p1Collection[[idx1]];   randomP2 = p2Collection[[idx2]];   (*计算位置差*)posDiff = Abs[idx1 - idx2];   (*创建颜色-位置差越小越蓝，越大越红*)hueValue = Rescale[posDiff, {0, 10}, {0.6, 0}];   lineColor = Blue;   (*绘制线段和端点*){Thickness[0.002], Opacity[0.7], lineColor,     Line[{randomP1, randomP2}]}, {30}]; (*循环30次*)   crTwoRandomP2Lines =   Table[(*随机选择p1集合中的索引*)   idx1 = RandomInteger[{1, Length[p2Collection]}];   (*计算p2集合中的有效索引范围*)minIdx2 = Max[1, idx1 - 6];   maxIdx2 = Min[Length[p2Collection], idx1 + 1];   (*在有效范围内随机选择p2索引*)idx2 = RandomInteger[{minIdx2, maxIdx2}];   (*获取点*)randomP1 = p2Collection[[idx1]];   randomP2 = p2Collection[[idx2]];   (*计算位置差*)posDiff = Abs[idx1 - idx2];   (*创建颜色-位置差越小越蓝，越大越红*)hueValue = Rescale[posDiff, {0, 10}, {0.6, 0}];   lineColor = Blue;   (*绘制线段和端点*){Thickness[0.002], Opacity[0.7], lineColor,     Line[{randomP1, randomP2}]}, {30}]; (*循环30次*) crTwoRandomP3P4Lines =   Table[(*随机选择p1集合中的索引*)   idx1 = RandomInteger[{1, Length[p3Collection]}];   (*计算p2集合中的有效索引范围*)minIdx2 = Max[1, idx1];   maxIdx2 = Min[Length[p4Collection], idx1 + 8];   (*在有效范围内随机选择p2索引*)idx2 = RandomInteger[{minIdx2, maxIdx2}];   (*获取点*)randomP1 = p3Collection[[idx1]];   randomP2 = p4Collection[[idx2]];   (*计算位置差*)posDiff = Abs[idx1 - idx2];   (*创建颜色-位置差越小越蓝，越大越红*)hueValue = Rescale[posDiff, {0, 10}, {0.6, 0}];   lineColor = Blue;   (*绘制线段和端点*){Thickness[0.002], Opacity[0.7], lineColor,     Line[{randomP1, randomP2}]}, {30}]; (*循环30次*)   png = Show[Graphics3D[crOneConnectors], Graphics3D[crOneConnectors1],   Graphics3D[crOneConnectors2], Graphics3D[crTwoConnectors],   Graphics3D[crTwoConnectors1], Graphics3D[crTwoConnectors2],   Graphics3D[crOneRandomP1P2Lines], Graphics3D[crOneRandomP2Lines],   Graphics3D[crOneRandomP3P4Lines], Graphics3D[crTwoRandomP1P2Lines],   Graphics3D[crTwoRandomP2Lines], Graphics3D[crTwoRandomP3P4Lines],   curve12,(*curve21,curve22,*)curve31, axes, Axes -> True,   AxesLabel -> {"X", "Y", "Z"},   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1},   ImageSize -> Large, PlotLabel -> "Bird's Nest"]
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最终效果：

gif-2025-11-21 at 22.54.40.gif
gif-2025-11-21 at 22.54.40.gif

借助 Ai 工具，一步步很容易实现最后的效果，也了解 Mathematica 在 3d 方面使用方法，但最终效果还是过于有规律，mark 一下，后续尝试在 manim 实现这个效果。

参考资源