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

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。

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。

icon

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

1curve = ParametricPlot3D[{4 Cos[t], 4.2 Sin[t], 2    0.6 Sin[2 t + Pi + Pi/2] + 3.2}, {t, 0, 2 Pi}, 3   PlotStyle -> {Thick, Blue}, 4   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];5 6axes = Graphics3D[{Arrowheads[0.05], {Red, 7     Arrow[{{0, 0, 0}, {5, 0, 0}}]},(*X 轴*){Green, 8     Arrow[{{0, 0, 0}, {0, 5, 0}}]},(*Y 轴*){Blue, 9     Arrow[{{0, 0, 0}, {0, 0, 5}}]}    (*Z 轴*)}];10 11 12png = Show[curve, axes, AxesLabel -> {"X", "Y", "Z"}, 13  LabelStyle -> {FontSize -> 14}, Boxed -> True]

mathematica 绘制鸟巢几条主要的轮廓线

Code

1curve12 = 2  ParametricPlot3D[{2 Cos[t], 1.5 Sin[t], 3    0.2 Sin[2 t + Pi + Pi/2] + 3}, {t, 0, 2 Pi}, 4   PlotStyle -> {Thick, Blue}, 5   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];6curve21 = 7  ParametricPlot3D[{4 Cos[t], 4.2 Sin[t], 8    0.6 Sin[2 t + Pi + Pi/2] + 3.2}, {t, 0, 2 Pi}, 9   PlotStyle -> {Thick, Blue}, 10   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];11curve22 = 12  ParametricPlot3D[{4.2 Cos[t], 4.4 Sin[t], 13    0.6 Sin[2 t + Pi + Pi/2] + 2.5}, {t, 0, 2 Pi}, 14   PlotStyle -> {Thick, Blue}, 15   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];16curve31 = 17  ParametricPlot3D[{3.4 Cos[t], 3.8 Sin[t], 0}, {t, 0, 2 \[Pi]}, 18   PlotStyle -> {Thick, Blue}, Axes -> False, 19   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];20 21axes = Graphics3D[{Arrowheads[0.05], {Red, 22     Arrow[{{0, 0, 0}, {5, 0, 0}}]},(*X 轴*){Green, 23     Arrow[{{0, 0, 0}, {0, 5, 0}}]},(*Y 轴*){Blue, 24     Arrow[{{0, 0, 0}, {0, 0, 5}}]}    (*Z 轴*)}];25 26 27png = Show[curve12, curve21, curve22, curve31, axes, 28  AxesLabel -> {"X", "Y", "Z"}, LabelStyle -> {FontSize -> 14}, 29  Boxed -> True]

mathematica 四段曲线上分别选取一个点，连接成线

Code

1curve12 = 2  ParametricPlot3D[{2 Cos[t], 1.5 Sin[t], 3    0.2 Sin[2 t + Pi + Pi/2] + 3}, {t, 0, 2 Pi}, 4   PlotStyle -> {Thick, Blue}, 5   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];6 7curve21 = 8  ParametricPlot3D[{4 Cos[t], 4.2 Sin[t], 9    0.6 Sin[2 t + Pi + Pi/2] + 3.2}, {t, 0, 2 Pi}, 10   PlotStyle -> {Thick, Blue}, 11   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];12 13curve22 = 14  ParametricPlot3D[{4.2 Cos[t], 4.4 Sin[t], 15    0.6 Sin[2 t + Pi + Pi/2] + 2.5}, {t, 0, 2 Pi}, 16   PlotStyle -> {Thick, Blue}, 17   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];18 19curve31 = 20  ParametricPlot3D[{3.4 Cos[t], 3.8 Sin[t], 0}, {t, 0, 2 \[Pi]}, 21   PlotStyle -> {Thick, Blue}, Axes -> False, 22   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];23 24axes = Graphics3D[{Arrowheads[0.05], {Red, 25     Arrow[{{0, 0, 0}, {5, 0, 0}}]},(*X 轴*){Green, 26     Arrow[{{0, 0, 0}, {0, 5, 0}}]},(*Y 轴*){Blue, 27     Arrow[{{0, 0, 0}, {0, 0, 5}}]}    (*Z 轴*)}];28 29 30p1 = {2 Cos[0], 1.5 Sin[0], 31   0.2 Sin[2*0 + Pi + Pi/2] + 3};       (*curve12 上的点*)32p2 = {4 Cos[0], 4.2 Sin[0], 33   0.6 Sin[2*0 + Pi + Pi/2] + 3.2};       (*curve21 上的点*)34p3 = {4.2 Cos[0], 4.4 Sin[0], 35   0.6 Sin[2*0 + Pi + Pi/2] + 2.5};   (*curve22 上的点*)36p4 = {3.4 Cos[0], 3.8 Sin[0], 37   0};                                (*curve31 上的点*)38 39(*连接四个点绘制红色直线*)40connector = 41  Graphics3D[{Thick, Red, Line[{p1, p2, p3, p4}],(*连接四个点的折线*)42    PointSize[0.02], {Black, Point[p1]},(*点1：黑色*){Black, 43     Point[p2]},(*点2：黑色*){Black, Point[p3]},(*点3：黑色*){Black, 44     Point[p4]}                 (*点4：黑色*)}];45 46 47png = Show[curve12, curve21, curve22, curve31, axes, connector, 48  AxesLabel -> {"X", "Y", "Z"}, LabelStyle -> {FontSize -> 14}, 49  Boxed -> True]

mathematica 四段曲线上分别选取一个点，直线替换为曲线

Code

1curve12 = 2  ParametricPlot3D[{2 Cos[t], 1.5 Sin[t], 3    0.2 Sin[2 t + Pi + Pi/2] + 3}, {t, 0, 2 Pi}, 4   PlotStyle -> {Thick, Blue}, 5   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];6curve21 = 7  ParametricPlot3D[{4 Cos[t], 4.2 Sin[t], 8    0.6 Sin[2 t + Pi + Pi/2] + 3.2}, {t, 0, 2 Pi}, 9   PlotStyle -> {Thick, Blue}, 10   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];11curve22 = 12  ParametricPlot3D[{4.2 Cos[t], 4.4 Sin[t], 13    0.6 Sin[2 t + Pi + Pi/2] + 2.5}, {t, 0, 2 Pi}, 14   PlotStyle -> {Thick, Blue}, 15   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];16curve31 = 17  ParametricPlot3D[{3.4 Cos[t], 3.8 Sin[t], 0}, {t, 0, 2 \[Pi]}, 18   PlotStyle -> {Thick, Blue}, Axes -> False, 19   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];20 21axes = Graphics3D[{Arrowheads[0.05], {Red, 22     Arrow[{{0, 0, 0}, {5, 0, 0}}]},(*X 轴*){Green, 23     Arrow[{{0, 0, 0}, {0, 5, 0}}]},(*Y 轴*){Blue, 24     Arrow[{{0, 0, 0}, {0, 0, 5}}]}    (*Z 轴*)}];25 26 27p1 = {2 Cos[0], 1.5 Sin[0], 28   0.2 Sin[2*0 + Pi + Pi/2] + 3};       (*curve12 上的点*)29p2 = {4 Cos[0], 4.2 Sin[0], 30   0.6 Sin[2*0 + Pi + Pi/2] + 3.2};       (*curve21 上的点*)31p3 = {4.2 Cos[0], 4.4 Sin[0], 32   0.6 Sin[2*0 + Pi + Pi/2] + 2.5};   (*curve22 上的点*)33p4 = {3.4 Cos[0], 3.8 Sin[0], 34   0};                                (*curve31 上的点*)35 36 37catmullRom3D[{p0_, p1_, p2_, p3_}, t_] := 38  0.5*{2 p1[[1]] + t (p2[[1]] - p0[[1]]) + 39     t^2 (2 p0[[1]] - 5 p1[[1]] + 4 p2[[1]] - p3[[1]]) + 40     t^3 (-p0[[1]] + 3 p1[[1]] - 3 p2[[1]] + p3[[1]]), 41    2 p1[[2]] + t (p2[[2]] - p0[[2]]) + 42     t^2 (2 p0[[2]] - 5 p1[[2]] + 4 p2[[2]] - p3[[2]]) + 43     t^3 (-p0[[2]] + 3 p1[[2]] - 3 p2[[2]] + p3[[2]]), 44    2 p1[[3]] + t (p2[[3]] - p0[[3]]) + 45     t^2 (2 p0[[3]] - 5 p1[[3]] + 4 p2[[3]] - p3[[3]]) + 46     t^3 (-p0[[3]] + 3 p1[[3]] - 3 p2[[3]] + p3[[3]])};47 48pts = {p1, p2, p3, p4};49newPts = 50  Join[{2.1 pts[[1]] - pts[[2]]}, pts, {2.2 pts[[-1]] - pts[[-2]]}];51 52curvePoints3D = 53  Table[catmullRom3D[{p1, p2, p3, p4}, t], {t, 0, 1, 0.1}];54 55curve = Graphics3D[{{Thick, Blue, Line[curvePoints3D]}, {Black, 56     PointSize[0.02], Point[{p1, p1, p2, p3}]}, {Dashed, Gray, 57     Line[{p1, p1, p2, p3}]}}];58 59 60png = Show[curve, curve12, curve21, curve22, curve31, axes, 61  AxesLabel -> {"X", "Y", "Z"}, LabelStyle -> {FontSize -> 14}, 62  Boxed -> True]

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

1curve12 = 2  ParametricPlot3D[{2 Cos[t], 1.5 Sin[t], 3    0.2 Sin[2 t + Pi + Pi/2] + 3}, {t, 0, 2 Pi}, 4   PlotStyle -> {Thick, Blue}, 5   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];6curve21 = 7  ParametricPlot3D[{4 Cos[t], 4.2 Sin[t], 8    0.6 Sin[2 t + Pi + Pi/2] + 3.2}, {t, 0, 2 Pi}, 9   PlotStyle -> {Thick, Blue}, 10   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];11curve22 = 12  ParametricPlot3D[{4.2 Cos[t], 4.4 Sin[t], 13    0.6 Sin[2 t + Pi + Pi/2] + 2.5}, {t, 0, 2 Pi}, 14   PlotStyle -> {Thick, Blue}, 15   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];16curve31 = 17  ParametricPlot3D[{3.4 Cos[t], 3.8 Sin[t], 0}, {t, 0, 2 \[Pi]}, 18   PlotStyle -> {Thick, Blue}, Axes -> False, 19   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];20 21axes = Graphics3D[{Arrowheads[0.05], {Red, 22     Arrow[{{0, 0, 0}, {5, 0, 0}}]},(*X 轴*){Green, 23     Arrow[{{0, 0, 0}, {0, 5, 0}}]},(*Y 轴*){Blue, 24     Arrow[{{0, 0, 0}, {0, 0, 5}}]}    (*Z 轴*)}];25 26p1 = {2 Cos[0], 1.5 Sin[0], 27   0.2 Sin[2*0 + Pi + Pi/2] + 3};       (*curve12 上的点*)28p2 = {4 Cos[0], 4.2 Sin[0], 29   0.6 Sin[2*0 + Pi + Pi/2] + 3.2};       (*curve21 上的点*)30p3 = {4.2 Cos[0], 4.4 Sin[0], 31   0.6 Sin[2*0 + Pi + Pi/2] + 2.5};   (*curve22 上的点*)32p4 = {3.4 Cos[0], 3.8 Sin[0], 33   0};                                (*curve31 上的点*)34 35catmullRom3D[{p0_, p1_, p2_, p3_}, t_] := 36  0.5*{2 p1[[1]] + t (p2[[1]] - p0[[1]]) + 37     t^2 (2 p0[[1]] - 5 p1[[1]] + 4 p2[[1]] - p3[[1]]) + 38     t^3 (-p0[[1]] + 3 p1[[1]] - 3 p2[[1]] + p3[[1]]), 39    2 p1[[2]] + t (p2[[2]] - p0[[2]]) + 40     t^2 (2 p0[[2]] - 5 p1[[2]] + 4 p2[[2]] - p3[[2]]) + 41     t^3 (-p0[[2]] + 3 p1[[2]] - 3 p2[[2]] + p3[[2]]), 42    2 p1[[3]] + t (p2[[3]] - p0[[3]]) + 43     t^2 (2 p0[[3]] - 5 p1[[3]] + 4 p2[[3]] - p3[[3]]) + 44     t^3 (-p0[[3]] + 3 p1[[3]] - 3 p2[[3]] + p3[[3]])};45 46pts = {p1, p2, p3, p4};47curvePoints3D = 48  Table[catmullRom3D[{p1, p2, p3, p4}, t], {t, 0, 1, 0.1}];49 50newPts = 51  Join[{2.1 pts[[1]] - pts[[2]]}, pts, {2.2 pts[[-1]] - pts[[-2]]}];52crPts = Flatten[53   Table[Table[54     catmullRom3D[newPts[[i ;; i + 3]], s], {s, 0, 1, 0.1}], {i, 1, 55     Length[newPts] - 3}], 1];56 57curve = Graphics3D[{{Thick, Blue, Line[crPts]}, {Black, 58     PointSize[0.02], Point[{p1, p1, p2, p3}]}, {Dashed, Gray, 59     Line[{p1, p1, p2, p3}]}}];60 61png = Show[curve, curve12, curve21, curve22, curve31, axes, 62  AxesLabel -> {"X", "Y", "Z"}, LabelStyle -> {FontSize -> 14}, 63  Boxed -> True]

mathematica 绘制一圈曲线

Code

1curve12 = 2  ParametricPlot3D[{2 Cos[t], 1.5 Sin[t], 3    0.2 Sin[2 t + Pi + Pi/2] + 3}, {t, 0, 2 Pi}, 4   PlotStyle -> {Thick, Blue}, 5   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];6curve21 = 7  ParametricPlot3D[{4 Cos[t], 4.2 Sin[t], 8    0.6 Sin[2 t + Pi + Pi/2] + 3.2}, {t, 0, 2 Pi}, 9   PlotStyle -> {Thick, Blue}, 10   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];11curve22 = 12  ParametricPlot3D[{4.2 Cos[t], 4.4 Sin[t], 13    0.6 Sin[2 t + Pi + Pi/2] + 2.5}, {t, 0, 2 Pi}, 14   PlotStyle -> {Thick, Blue}, 15   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];16curve31 = 17  ParametricPlot3D[{3.4 Cos[t], 3.8 Sin[t], 0}, {t, 0, 2 \[Pi]}, 18   PlotStyle -> {Thick, Blue}, Axes -> False, 19   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];20 21axes = Graphics3D[{Arrowheads[0.05], {Red, 22     Arrow[{{0, 0, 0}, {5, 0, 0}}]},(*X 轴*){Green, 23     Arrow[{{0, 0, 0}, {0, 5, 0}}]},(*Y 轴*){Blue, 24     Arrow[{{0, 0, 0}, {0, 0, 5}}]}    (*Z 轴*)}];25 26catmullRom3D[{p0_, p1_, p2_, p3_}, t_] := 27  0.5*{2 p1[[1]] + t (p2[[1]] - p0[[1]]) + 28     t^2 (2 p0[[1]] - 5 p1[[1]] + 4 p2[[1]] - p3[[1]]) + 29     t^3 (-p0[[1]] + 3 p1[[1]] - 3 p2[[1]] + p3[[1]]), 30    2 p1[[2]] + t (p2[[2]] - p0[[2]]) + 31     t^2 (2 p0[[2]] - 5 p1[[2]] + 4 p2[[2]] - p3[[2]]) + 32     t^3 (-p0[[2]] + 3 p1[[2]] - 3 p2[[2]] + p3[[2]]), 33    2 p1[[3]] + t (p2[[3]] - p0[[3]]) + 34     t^2 (2 p0[[3]] - 5 p1[[3]] + 4 p2[[3]] - p3[[3]]) + 35     t^3 (-p0[[3]] + 3 p1[[3]] - 3 p2[[3]] + p3[[3]])};36 37 38offset = 0;39step = 0.2540crOneOffsets = {0 + offset, -Pi/16 + offset, -Pi/28 + offset, 41   Pi/8 + offset};42crOneTs = 43  Range[0, 2 Pi - step, 44   step];                            (*curve31 上的点*)45crOneConnectors = Table[random = RandomReal[{1, 15}]*0.02;46   p1 = {2 Cos[t + crOneOffsets[[1]] + random], 47     1.5 Sin[t + crOneOffsets[[1]] + random], 48     0.2 Sin[2 (t + crOneOffsets[[1]]) + 3 Pi/2] + 3};49   p2 = {4 Cos[t + crOneOffsets[[2]]], 4.2 Sin[t + crOneOffsets[[2]]],50      0.6 Sin[2 (t + crOneOffsets[[2]]) + 3 Pi/2] + 3.2};51   p3 = {4.2 Cos[t + crOneOffsets[[3]]], 52     4.4 Sin[t + crOneOffsets[[3]]], 53     0.6 Sin[2 (t + crOneOffsets[[3]]) + 3 Pi/2] + 2.5};54   p4 = {3.4 Cos[t + crOneOffsets[[4]] + random], 55     3.8 Sin[t + crOneOffsets[[4]] + random], 0};56   pts = {p1, p2, p3, p4};57   newPts = 58    Join[{2.1 pts[[1]] - pts[[2]]}, 59     pts, {2.2 pts[[-1]] - pts[[-2]]}];60   crPts = 61    Flatten[Table[62      Table[catmullRom3D[newPts[[i ;; i + 3]], s], {s, 0, 1, 63        1/30}], {i, 1, Length[newPts] - 3}], 1];64   (*4. 绘制这一段的 CR 曲线+原控制点*){Thick, Opacity[0.7], Blue, Line[crPts], 65    Black, PointSize[0.015], Point[pts]}, {t, crOneTs}];66 67 68png = Show[Graphics3D[crOneConnectors], curve12, curve21, curve22, 69  curve31, axes, Axes -> True, AxesLabel -> {"X", "Y", "Z"}, 70  PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}, 71  ImageSize -> Large, PlotLabel -> "Bird's Nest"]

mathematica 以此类推，绘制更多

Code

1curve12 = 2  ParametricPlot3D[{2 Cos[t], 1.5 Sin[t], 3    0.2 Sin[2 t + Pi + Pi/2] + 3}, {t, 0, 2 Pi}, 4   PlotStyle -> {Thick, Blue}, 5   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];6curve21 = 7  ParametricPlot3D[{4 Cos[t], 4.2 Sin[t], 8    0.6 Sin[2 t + Pi + Pi/2] + 3.2}, {t, 0, 2 Pi}, 9   PlotStyle -> {Thick, Blue}, 10   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];11curve22 = 12  ParametricPlot3D[{4.2 Cos[t], 4.4 Sin[t], 13    0.6 Sin[2 t + Pi + Pi/2] + 2.5}, {t, 0, 2 Pi}, 14   PlotStyle -> {Thick, Blue}, 15   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];16curve31 = 17  ParametricPlot3D[{3.4 Cos[t], 3.8 Sin[t], 0}, {t, 0, 2 \[Pi]}, 18   PlotStyle -> {Thick, Blue}, Axes -> False, 19   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];20 21axes = Graphics3D[{Arrowheads[0.05], {Red, 22     Arrow[{{0, 0, 0}, {5, 0, 0}}]},(*X 轴*){Green, 23     Arrow[{{0, 0, 0}, {0, 5, 0}}]},(*Y 轴*){Blue, 24     Arrow[{{0, 0, 0}, {0, 0, 5}}]}    (*Z 轴*)}];25 26catmullRom3D[{p0_, p1_, p2_, p3_}, t_] := 27  0.5*{2 p1[[1]] + t (p2[[1]] - p0[[1]]) + 28     t^2 (2 p0[[1]] - 5 p1[[1]] + 4 p2[[1]] - p3[[1]]) + 29     t^3 (-p0[[1]] + 3 p1[[1]] - 3 p2[[1]] + p3[[1]]), 30    2 p1[[2]] + t (p2[[2]] - p0[[2]]) + 31     t^2 (2 p0[[2]] - 5 p1[[2]] + 4 p2[[2]] - p3[[2]]) + 32     t^3 (-p0[[2]] + 3 p1[[2]] - 3 p2[[2]] + p3[[2]]), 33    2 p1[[3]] + t (p2[[3]] - p0[[3]]) + 34     t^2 (2 p0[[3]] - 5 p1[[3]] + 4 p2[[3]] - p3[[3]]) + 35     t^3 (-p0[[3]] + 3 p1[[3]] - 3 p2[[3]] + p3[[3]])};36 37 38offset = 0;39step = 0.2540crOneOffsets = {0 + offset, -Pi/16 + offset, -Pi/28 + offset, 41   Pi/8 + offset};42crOneTs = 43  Range[0, 2 Pi - step, 44   step];                            (*curve31 上的点*)45crOneConnectors = Table[random = RandomReal[{1, 15}]*0.02;46   p1 = {2 Cos[t + crOneOffsets[[1]] + random], 47     1.5 Sin[t + crOneOffsets[[1]] + random], 48     0.2 Sin[2 (t + crOneOffsets[[1]]) + 3 Pi/2] + 3};49   p2 = {4 Cos[t + crOneOffsets[[2]]], 4.2 Sin[t + crOneOffsets[[2]]],50      0.6 Sin[2 (t + crOneOffsets[[2]]) + 3 Pi/2] + 3.2};51   p3 = {4.2 Cos[t + crOneOffsets[[3]]], 52     4.4 Sin[t + crOneOffsets[[3]]], 53     0.6 Sin[2 (t + crOneOffsets[[3]]) + 3 Pi/2] + 2.5};54   p4 = {3.4 Cos[t + crOneOffsets[[4]] + random], 55     3.8 Sin[t + crOneOffsets[[4]] + random], 0};56   pts = {p1, p2, p3, p4};57   newPts = 58    Join[{2.1 pts[[1]] - pts[[2]]}, 59     pts, {2.2 pts[[-1]] - pts[[-2]]}];60   crPts = 61    Flatten[Table[62      Table[catmullRom3D[newPts[[i ;; i + 3]], s], {s, 0, 1, 63        1/30}], {i, 1, Length[newPts] - 3}], 1];64   (*4. 绘制这一段的 CR 曲线+原控制点*){Thick, Opacity[0.7], Blue, Line[crPts], 65    Black, PointSize[0.015], Point[pts]}, {t, crOneTs}];66 67crOneConnectors1 = Table[random = RandomReal[{1, 15}]*0.02;68   p1 = {2 Cos[t + crOneOffsets[[1]] + random], 69     1.5 Sin[t + crOneOffsets[[1]] + random], 70     0.2 Sin[2 (t + crOneOffsets[[1]] + random) + 3 Pi/2] + 3};71   p2 = {4 Cos[t + crOneOffsets[[2]]], 4.2 Sin[t + crOneOffsets[[2]]],72      0.6 Sin[2 (t + crOneOffsets[[2]]) + 3 Pi/2] + 3.2};73   p3 = {4.2 Cos[t + crOneOffsets[[3]] + step], 74     4.4 Sin[t + crOneOffsets[[3]] + step], 75     0.6 Sin[2 (t + crOneOffsets[[3]] + step) + 3 Pi/2] + 2.5};76   p4 = {3.4 Cos[t + crOneOffsets[[4]] + random + step], 77     3.8 Sin[t + crOneOffsets[[4]] + random + step], 0};78   pts = {p1, p2, p3, p4};79   crPts = 80    Flatten[Table[81      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i, 82       1, Length[pts] - 3}], 1];83   (*4. 绘制这一段的 CR 曲线+原控制点*){Thick, Opacity[0.7], Blue, Line[crPts], 84    Transparent, PointSize[0.001], Point[pts]}, {t, crOneTs}];85 86crOneConnectors2 = Table[random = RandomReal[{1, 15}]*0.02;87   p1 = {2 Cos[t + crOneOffsets[[1]] + random], 88     1.5 Sin[t + crOneOffsets[[1]] + random], 89     0.2 Sin[2 (t + crOneOffsets[[1]] + random) + 3 Pi/2] + 3};90   p2 = {4 Cos[t + crOneOffsets[[2]] + step], 91     4.2 Sin[t + crOneOffsets[[2]] + step], 92     0.6 Sin[2 (t + crOneOffsets[[2]] + step) + 3 Pi/2] + 3.2};93   p3 = {4.2 Cos[t + crOneOffsets[[3]]], 94     4.4 Sin[t + crOneOffsets[[3]]], 95     0.6 Sin[2 (t + crOneOffsets[[3]]) + 3 Pi/2] + 2.5};96   p4 = {3.4 Cos[t + crOneOffsets[[4]] + random - step], 97     3.8 Sin[t + crOneOffsets[[4]] + random - step], 0};98   pts = {p1, p2, p3, p4};99   crPts = 100    Flatten[Table[101      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i, 102       1, Length[pts] - 3}], 1];103   (*4. 绘制这一段的 CR 曲线+原控制点*){Thick, Opacity[0.7], Blue, Line[crPts], 104    Transparent, PointSize[0.001], Point[pts]}, {t, crOneTs}];105 106 107offset = Pi/32;108crTwoOffsets = {0 + offset, Pi/16 + offset, 109   Pi/28 + offset, -Pi/8 + offset};110crTwoTs = Range[0, 2 Pi, step];111crTwoConnectors = Table[random = RandomReal[{1, 15}]*0.02;112   p1 = {2 Cos[t + crTwoOffsets[[1]] + random], 113     1.5 Sin[t + crTwoOffsets[[1]] + random], 114     0.2 Sin[2 (t + crTwoOffsets[[1]]) + 3 Pi/2] + 3};115   p2 = {4 Cos[t + crTwoOffsets[[2]]], 4.2 Sin[t + crTwoOffsets[[2]]],116      0.6 Sin[2 (t + crTwoOffsets[[2]]) + 3 Pi/2] + 3.2};117   p3 = {4.2 Cos[t + crTwoOffsets[[3]]], 118     4.4 Sin[t + crTwoOffsets[[3]]], 119     0.6 Sin[2 (t + crTwoOffsets[[3]]) + 3 Pi/2] + 2.5};120   p4 = {3.4 Cos[t + crTwoOffsets[[4]] + random], 121     3.8 Sin[t + crTwoOffsets[[4]] + random], 0};122   pts = {p1, p2, p3, p4};123   newPts = 124    Join[{2.1 pts[[1]] - pts[[2]]},(*p0=2*p1-p2*)125     pts, {2.2 pts[[-1]] - pts[[-2]]}];126   crPts = 127    Flatten[Table[128      Table[catmullRom3D[newPts[[i ;; i + 3]], s], {s, 0, 1, 129        1/30}], {i, 1, Length[newPts] - 3}], 1];130   {Thick, Opacity[0.7], Blue, Line[crPts], Transparent, 131    PointSize[0.001], Point[pts]}, {t, crTwoTs}];132 133crTwoConnectors1 = Table[random = RandomReal[{1, 15}]*0.02;134   p1 = {2 Cos[t + crTwoOffsets[[1]] + random], 135     1.5 Sin[t + crTwoOffsets[[1]] + random], 136     0.2 Sin[2 (t + crTwoOffsets[[1]]) + 3 Pi/2] + 3};137   p2 = {4 Cos[t + crTwoOffsets[[2]]], 4.2 Sin[t + crTwoOffsets[[2]]],138      0.6 Sin[2 (t + crTwoOffsets[[2]]) + 3 Pi/2] + 3.2};139   p3 = {4.2 Cos[t + crTwoOffsets[[3]] + step], 140     4.4 Sin[t + crTwoOffsets[[3]] + step], 141     0.6 Sin[2 (t + crTwoOffsets[[3]] + step) + 3 Pi/2] + 2.5};142   p4 = {3.4 Cos[t + crTwoOffsets[[4]] + random + step], 143     3.8 Sin[t + crTwoOffsets[[4]] + random + step], 0};144   pts = {p1, p2, p3, p4};145   crPts = 146    Flatten[Table[147      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i, 148       1, Length[pts] - 3}], 1];149   {Thick, Opacity[0.7], Blue, Line[crPts], Transparent, 150    PointSize[0.001], Point[pts]}, {t, crTwoTs}];151 152 153crTwoConnectors2 = Table[random = RandomReal[{1, 15}]*0.02;154   p1 = {2 Cos[t + crTwoOffsets[[1]] + random], 155     1.5 Sin[t + crTwoOffsets[[1]] + random], 156     0.2 Sin[2 (t + crTwoOffsets[[1]]) + 3 Pi/2] + 3};157   p2 = {4 Cos[t + crTwoOffsets[[2]] + step], 158     4.2 Sin[t + crTwoOffsets[[2]] + step], 159     0.6 Sin[2 (t + crTwoOffsets[[2]] + step) + 3 Pi/2] + 3.2};160   p3 = {4.2 Cos[t + crTwoOffsets[[3]]], 161     4.4 Sin[t + crTwoOffsets[[3]]], 162     0.6 Sin[2 (t + crTwoOffsets[[3]]) + 3 Pi/2] + 2.5};163   p4 = {3.4 Cos[t + crTwoOffsets[[4]] + random - step], 164     3.8 Sin[t + crTwoOffsets[[4]] + random - step], 0};165   pts = {p1, p2, p3, p4};166   crPts = 167    Flatten[Table[168      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i, 169       1, Length[pts] - 3}], 1];170   {Thick, Opacity[0.7], Blue, Line[crPts], Transparent, 171    PointSize[0.001], Point[pts]}, {t, crTwoTs}];172 173 174 175 176 177 178png = Show[Graphics3D[crOneConnectors], Graphics3D[crOneConnectors1], 179  Graphics3D[crOneConnectors2], Graphics3D[crTwoConnectors], 180  Graphics3D[crTwoConnectors1], Graphics3D[crTwoConnectors2], curve12,181   curve21, curve22, curve31, axes, Axes -> True, 182  AxesLabel -> {"X", "Y", "Z"}, 183  PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}, 184  ImageSize -> Large, PlotLabel -> "Bird's Nest"]

mathematica 调整下图形

Code

1curve12 = 2  ParametricPlot3D[{2 Cos[t], 1.5 Sin[t], 3    0.2 Sin[2 t + Pi + Pi/2] + 3}, {t, 0, 2 Pi}, 4   PlotStyle -> {Thickness[0.002], Blue}, 5   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];6curve21 = 7  ParametricPlot3D[{4 Cos[t], 4.2 Sin[t], 8    0.6 Sin[2 t + Pi + Pi/2] + 3.2}, {t, 0, 2 Pi}, 9   PlotStyle -> {Thickness[0.002], Blue}, 10   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];11curve22 = 12  ParametricPlot3D[{4.2 Cos[t], 4.4 Sin[t], 13    0.6 Sin[2 t + Pi + Pi/2] + 2.5}, {t, 0, 2 Pi}, 14   PlotStyle -> {Thickness[0.002], Blue}, 15   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];16curve31 = 17  ParametricPlot3D[{3.4 Cos[t], 3.8 Sin[t], 0}, {t, 0, 2 \[Pi]}, 18   PlotStyle -> {Thickness[0.002], Blue}, Axes -> False, 19   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];20 21axes = Graphics3D[{Arrowheads[0.05], {Red, 22     Arrow[{{0, 0, 0}, {5, 0, 0}}]},(*X 轴*){Green, 23     Arrow[{{0, 0, 0}, {0, 5, 0}}]},(*Y 轴*){Blue, 24     Arrow[{{0, 0, 0}, {0, 0, 5}}]}    (*Z 轴*)}];25 26catmullRom3D[{p0_, p1_, p2_, p3_}, t_] := 27  0.5*{2 p1[[1]] + t (p2[[1]] - p0[[1]]) + 28     t^2 (2 p0[[1]] - 5 p1[[1]] + 4 p2[[1]] - p3[[1]]) + 29     t^3 (-p0[[1]] + 3 p1[[1]] - 3 p2[[1]] + p3[[1]]), 30    2 p1[[2]] + t (p2[[2]] - p0[[2]]) + 31     t^2 (2 p0[[2]] - 5 p1[[2]] + 4 p2[[2]] - p3[[2]]) + 32     t^3 (-p0[[2]] + 3 p1[[2]] - 3 p2[[2]] + p3[[2]]), 33    2 p1[[3]] + t (p2[[3]] - p0[[3]]) + 34     t^2 (2 p0[[3]] - 5 p1[[3]] + 4 p2[[3]] - p3[[3]]) + 35     t^3 (-p0[[3]] + 3 p1[[3]] - 3 p2[[3]] + p3[[3]])};36 37 38offset = 0;39step = 0.2540crOneOffsets = {0 + offset, -Pi/16 + offset, -Pi/28 + offset, 41   Pi/8 + offset};42crOneTs = 43  Range[0, 2 Pi - step, 44   step];                            (*curve31 上的点*)45crOneConnectors = Table[random = RandomReal[{1, 15}]*0.02;46   p1 = {2 Cos[t + crOneOffsets[[1]] + random], 47     1.5 Sin[t + crOneOffsets[[1]] + random], 48     0.2 Sin[2 (t + crOneOffsets[[1]]) + 3 Pi/2] + 3};49   p2 = {4 Cos[t + crOneOffsets[[2]]], 4.2 Sin[t + crOneOffsets[[2]]],50      0.6 Sin[2 (t + crOneOffsets[[2]]) + 3 Pi/2] + 3.2};51   p3 = {4.2 Cos[t + crOneOffsets[[3]]], 52     4.4 Sin[t + crOneOffsets[[3]]], 53     0.6 Sin[2 (t + crOneOffsets[[3]]) + 3 Pi/2] + 2.5};54   p4 = {3.4 Cos[t + crOneOffsets[[4]] + random], 55     3.8 Sin[t + crOneOffsets[[4]] + random], 0};56   pts = {p1, p2, p3, p4};57   newPts = 58    Join[{2.1 pts[[1]] - pts[[2]]}, 59     pts, {2.2 pts[[-1]] - pts[[-2]]}];60   crPts = 61    Flatten[Table[62      Table[catmullRom3D[newPts[[i ;; i + 3]], s], {s, 0, 1, 63        1/30}], {i, 1, Length[newPts] - 3}], 1];64   (*4. 绘制这一段的 CR 曲线+原控制点*){Thickness[0.002], Opacity[0.7], Blue, 65    Line[crPts], Transparent, PointSize[0.015], Point[pts]}, {t, 66    crOneTs}];67 68crOneConnectors1 = Table[random = RandomReal[{1, 15}]*0.02;69   p1 = {2 Cos[t + crOneOffsets[[1]] + random], 70     1.5 Sin[t + crOneOffsets[[1]] + random], 71     0.2 Sin[2 (t + crOneOffsets[[1]] + random) + 3 Pi/2] + 3};72   p2 = {4 Cos[t + crOneOffsets[[2]]], 4.2 Sin[t + crOneOffsets[[2]]],73      0.6 Sin[2 (t + crOneOffsets[[2]]) + 3 Pi/2] + 3.2};74   p3 = {4.2 Cos[t + crOneOffsets[[3]] + step], 75     4.4 Sin[t + crOneOffsets[[3]] + step], 76     0.6 Sin[2 (t + crOneOffsets[[3]] + step) + 3 Pi/2] + 2.5};77   p4 = {3.4 Cos[t + crOneOffsets[[4]] + random + step], 78     3.8 Sin[t + crOneOffsets[[4]] + random + step], 0};79   pts = {p1, p2, p3, p4};80   crPts = 81    Flatten[Table[82      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i, 83       1, Length[pts] - 3}], 1];84   (*4. 绘制这一段的 CR 曲线+原控制点*){Thickness[0.002], Opacity[0.7], Blue, 85    Line[crPts], Transparent, PointSize[0.001], Point[pts]}, {t, 86    crOneTs}];87 88crOneConnectors2 = Table[random = RandomReal[{1, 15}]*0.02;89   p1 = {2 Cos[t + crOneOffsets[[1]] + random], 90     1.5 Sin[t + crOneOffsets[[1]] + random], 91     0.2 Sin[2 (t + crOneOffsets[[1]] + random) + 3 Pi/2] + 3};92   p2 = {4 Cos[t + crOneOffsets[[2]] + step], 93     4.2 Sin[t + crOneOffsets[[2]] + step], 94     0.6 Sin[2 (t + crOneOffsets[[2]] + step) + 3 Pi/2] + 3.2};95   p3 = {4.2 Cos[t + crOneOffsets[[3]]], 96     4.4 Sin[t + crOneOffsets[[3]]], 97     0.6 Sin[2 (t + crOneOffsets[[3]]) + 3 Pi/2] + 2.5};98   p4 = {3.4 Cos[t + crOneOffsets[[4]] + random - step], 99     3.8 Sin[t + crOneOffsets[[4]] + random - step], 0};100   pts = {p1, p2, p3, p4};101   crPts = 102    Flatten[Table[103      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i, 104       1, Length[pts] - 3}], 1];105   (*4. 绘制这一段的 CR 曲线+原控制点*){Thickness[0.002], Opacity[0.7], Blue, 106    Line[crPts], Transparent, PointSize[0.001], Point[pts]}, {t, 107    crOneTs}];108 109 110offset = Pi/32;111crTwoOffsets = {0 + offset, Pi/16 + offset, 112   Pi/28 + offset, -Pi/8 + offset};113crTwoTs = Range[0, 2 Pi, step];114crTwoConnectors = Table[random = RandomReal[{1, 15}]*0.02;115   p1 = {2 Cos[t + crTwoOffsets[[1]] + random], 116     1.5 Sin[t + crTwoOffsets[[1]] + random], 117     0.2 Sin[2 (t + crTwoOffsets[[1]]) + 3 Pi/2] + 3};118   p2 = {4 Cos[t + crTwoOffsets[[2]]], 4.2 Sin[t + crTwoOffsets[[2]]],119      0.6 Sin[2 (t + crTwoOffsets[[2]]) + 3 Pi/2] + 3.2};120   p3 = {4.2 Cos[t + crTwoOffsets[[3]]], 121     4.4 Sin[t + crTwoOffsets[[3]]], 122     0.6 Sin[2 (t + crTwoOffsets[[3]]) + 3 Pi/2] + 2.5};123   p4 = {3.4 Cos[t + crTwoOffsets[[4]] + random], 124     3.8 Sin[t + crTwoOffsets[[4]] + random], 0};125   pts = {p1, p2, p3, p4};126   newPts = 127    Join[{2.1 pts[[1]] - pts[[2]]},(*p0=2*p1-p2*)128     pts, {2.2 pts[[-1]] - pts[[-2]]}];129   crPts = 130    Flatten[Table[131      Table[catmullRom3D[newPts[[i ;; i + 3]], s], {s, 0, 1, 132        1/30}], {i, 1, Length[newPts] - 3}], 1];133   {Thickness[0.002], Opacity[0.7], Blue, Line[crPts], Transparent, 134    PointSize[0.001], Point[pts]}, {t, crTwoTs}];135 136crTwoConnectors1 = Table[random = RandomReal[{1, 15}]*0.02;137   p1 = {2 Cos[t + crTwoOffsets[[1]] + random], 138     1.5 Sin[t + crTwoOffsets[[1]] + random], 139     0.2 Sin[2 (t + crTwoOffsets[[1]]) + 3 Pi/2] + 3};140   p2 = {4 Cos[t + crTwoOffsets[[2]]], 4.2 Sin[t + crTwoOffsets[[2]]],141      0.6 Sin[2 (t + crTwoOffsets[[2]]) + 3 Pi/2] + 3.2};142   p3 = {4.2 Cos[t + crTwoOffsets[[3]] + step], 143     4.4 Sin[t + crTwoOffsets[[3]] + step], 144     0.6 Sin[2 (t + crTwoOffsets[[3]] + step) + 3 Pi/2] + 2.5};145   p4 = {3.4 Cos[t + crTwoOffsets[[4]] + random + step], 146     3.8 Sin[t + crTwoOffsets[[4]] + random + step], 0};147   pts = {p1, p2, p3, p4};148   crPts = 149    Flatten[Table[150      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i, 151       1, Length[pts] - 3}], 1];152   {Thickness[0.002], Opacity[0.7], Blue, Line[crPts], Transparent, 153    PointSize[0.001], Point[pts]}, {t, crTwoTs}];154 155 156crTwoConnectors2 = Table[random = RandomReal[{1, 15}]*0.02;157   p1 = {2 Cos[t + crTwoOffsets[[1]] + random], 158     1.5 Sin[t + crTwoOffsets[[1]] + random], 159     0.2 Sin[2 (t + crTwoOffsets[[1]]) + 3 Pi/2] + 3};160   p2 = {4 Cos[t + crTwoOffsets[[2]] + step], 161     4.2 Sin[t + crTwoOffsets[[2]] + step], 162     0.6 Sin[2 (t + crTwoOffsets[[2]] + step) + 3 Pi/2] + 3.2};163   p3 = {4.2 Cos[t + crTwoOffsets[[3]]], 164     4.4 Sin[t + crTwoOffsets[[3]]], 165     0.6 Sin[2 (t + crTwoOffsets[[3]]) + 3 Pi/2] + 2.5};166   p4 = {3.4 Cos[t + crTwoOffsets[[4]] + random - step], 167     3.8 Sin[t + crTwoOffsets[[4]] + random - step], 0};168   pts = {p1, p2, p3, p4};169   crPts = 170    Flatten[Table[171      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i, 172       1, Length[pts] - 3}], 1];173   {Thickness[0.002], Opacity[0.7], Blue, Line[crPts], Transparent, 174    PointSize[0.001], Point[pts]}, {t, crTwoTs}];175 176 177 178 179png = Show[Graphics3D[crOneConnectors], Graphics3D[crOneConnectors1], 180  Graphics3D[crOneConnectors2], Graphics3D[crTwoConnectors], 181  Graphics3D[crTwoConnectors1], Graphics3D[crTwoConnectors2], 182  curve12,(*curve21,curve22,*)curve31, axes, Axes -> True, 183  AxesLabel -> {"X", "Y", "Z"}, 184  PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}, 185  ImageSize -> Large, PlotLabel -> "Bird's Nest"]

mathematica 随机添加一些横线

Code

1curve12 = 2  ParametricPlot3D[{2 Cos[t], 1.5 Sin[t], 3    0.2 Sin[2 t + Pi + Pi/2] + 3}, {t, 0, 2 Pi}, 4   PlotStyle -> {Thickness[0.002], Blue}, 5   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];6curve21 = 7  ParametricPlot3D[{4 Cos[t], 4.2 Sin[t], 8    0.6 Sin[2 t + Pi + Pi/2] + 3.2}, {t, 0, 2 Pi}, 9   PlotStyle -> {Thickness[0.002], Blue}, 10   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];11curve22 = 12  ParametricPlot3D[{4.2 Cos[t], 4.4 Sin[t], 13    0.6 Sin[2 t + Pi + Pi/2] + 2.5}, {t, 0, 2 Pi}, 14   PlotStyle -> {Thickness[0.002], Blue}, 15   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];16curve31 = 17  ParametricPlot3D[{3.4 Cos[t], 3.8 Sin[t], 0}, {t, 0, 2 \[Pi]}, 18   PlotStyle -> {Thickness[0.002], Blue}, Axes -> False, 19   PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}];20 21axes = Graphics3D[{Arrowheads[0.05], {Red, 22     Arrow[{{0, 0, 0}, {5, 0, 0}}]},(*X 轴*){Green, 23     Arrow[{{0, 0, 0}, {0, 5, 0}}]},(*Y 轴*){Blue, 24     Arrow[{{0, 0, 0}, {0, 0, 5}}]}    (*Z 轴*)}];25 26catmullRom3D[{p0_, p1_, p2_, p3_}, t_] := 27  0.5*{2 p1[[1]] + t (p2[[1]] - p0[[1]]) + 28     t^2 (2 p0[[1]] - 5 p1[[1]] + 4 p2[[1]] - p3[[1]]) + 29     t^3 (-p0[[1]] + 3 p1[[1]] - 3 p2[[1]] + p3[[1]]), 30    2 p1[[2]] + t (p2[[2]] - p0[[2]]) + 31     t^2 (2 p0[[2]] - 5 p1[[2]] + 4 p2[[2]] - p3[[2]]) + 32     t^3 (-p0[[2]] + 3 p1[[2]] - 3 p2[[2]] + p3[[2]]), 33    2 p1[[3]] + t (p2[[3]] - p0[[3]]) + 34     t^2 (2 p0[[3]] - 5 p1[[3]] + 4 p2[[3]] - p3[[3]]) + 35     t^3 (-p0[[3]] + 3 p1[[3]] - 3 p2[[3]] + p3[[3]])};36 37p1Collection = {};38p2Collection = {};39p3Collection = {};40p4Collection = {};41 42offset = 0;43step = 0.2544crOneOffsets = {0 + offset, -Pi/16 + offset, -Pi/28 + offset, 45   Pi/8 + offset};46crOneTs = 47  Range[0, 2 Pi - step, 48   step];                            (*curve31 上的点*)49crOneConnectors = Table[random = RandomReal[{1, 15}]*0.02;50   p1 = {2 Cos[t + crOneOffsets[[1]] + random], 51     1.5 Sin[t + crOneOffsets[[1]] + random], 52     0.2 Sin[2 (t + crOneOffsets[[1]]) + 3 Pi/2] + 3};53   p2 = {4 Cos[t + crOneOffsets[[2]]], 4.2 Sin[t + crOneOffsets[[2]]],54      0.6 Sin[2 (t + crOneOffsets[[2]]) + 3 Pi/2] + 3.2};55   p3 = {4.2 Cos[t + crOneOffsets[[3]]], 56     4.4 Sin[t + crOneOffsets[[3]]], 57     0.6 Sin[2 (t + crOneOffsets[[3]]) + 3 Pi/2] + 2.5};58   p4 = {3.4 Cos[t + crOneOffsets[[4]] + random], 59     3.8 Sin[t + crOneOffsets[[4]] + random], 0};60   AppendTo[p1Collection, p1];61   AppendTo[p2Collection, p2];62   AppendTo[p3Collection, p3];63   AppendTo[p4Collection, p4];64   pts = {p1, p2, p3, p4};65   newPts = 66    Join[{2.1 pts[[1]] - pts[[2]]}, 67     pts, {2.2 pts[[-1]] - pts[[-2]]}];68   crPts = 69    Flatten[Table[70      Table[catmullRom3D[newPts[[i ;; i + 3]], s], {s, 0, 1, 71        1/30}], {i, 1, Length[newPts] - 3}], 1];72   (*4. 绘制这一段的 CR 曲线+原控制点*){Thickness[0.002], Opacity[0.7], Blue, 73    Line[crPts], Transparent, PointSize[0.015], Point[pts]}, {t, 74    crOneTs}];75 76crOneConnectors1 = Table[random = RandomReal[{1, 15}]*0.02;77   p1 = {2 Cos[t + crOneOffsets[[1]] + random], 78     1.5 Sin[t + crOneOffsets[[1]] + random], 79     0.2 Sin[2 (t + crOneOffsets[[1]] + random) + 3 Pi/2] + 3};80   p2 = {4 Cos[t + crOneOffsets[[2]]], 4.2 Sin[t + crOneOffsets[[2]]],81      0.6 Sin[2 (t + crOneOffsets[[2]]) + 3 Pi/2] + 3.2};82   p3 = {4.2 Cos[t + crOneOffsets[[3]] + step], 83     4.4 Sin[t + crOneOffsets[[3]] + step], 84     0.6 Sin[2 (t + crOneOffsets[[3]] + step) + 3 Pi/2] + 2.5};85   p4 = {3.4 Cos[t + crOneOffsets[[4]] + random + step], 86     3.8 Sin[t + crOneOffsets[[4]] + random + step], 0};87   pts = {p1, p2, p3, p4};88   crPts = 89    Flatten[Table[90      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i, 91       1, Length[pts] - 3}], 1];92   (*4. 绘制这一段的 CR 曲线+原控制点*){Thickness[0.002], Opacity[0.7], Blue, 93    Line[crPts], Transparent, PointSize[0.001], Point[pts]}, {t, 94    crOneTs}];95 96crOneConnectors2 = Table[random = RandomReal[{1, 15}]*0.02;97   p1 = {2 Cos[t + crOneOffsets[[1]] + random], 98     1.5 Sin[t + crOneOffsets[[1]] + random], 99     0.2 Sin[2 (t + crOneOffsets[[1]] + random) + 3 Pi/2] + 3};100   p2 = {4 Cos[t + crOneOffsets[[2]] + step], 101     4.2 Sin[t + crOneOffsets[[2]] + step], 102     0.6 Sin[2 (t + crOneOffsets[[2]] + step) + 3 Pi/2] + 3.2};103   p3 = {4.2 Cos[t + crOneOffsets[[3]]], 104     4.4 Sin[t + crOneOffsets[[3]]], 105     0.6 Sin[2 (t + crOneOffsets[[3]]) + 3 Pi/2] + 2.5};106   p4 = {3.4 Cos[t + crOneOffsets[[4]] + random - step], 107     3.8 Sin[t + crOneOffsets[[4]] + random - step], 0};108   pts = {p1, p2, p3, p4};109   crPts = 110    Flatten[Table[111      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i, 112       1, Length[pts] - 3}], 1];113   (*4. 绘制这一段的 CR 曲线+原控制点*){Thickness[0.002], Opacity[0.7], Blue, 114    Line[crPts], Transparent, PointSize[0.001], Point[pts]}, {t, 115    crOneTs}];116 117(*生成30条随机连接线，确保位置差不超过30*)118crOneRandomP1P2Lines = 119  Table[(*随机选择p1集合中的索引*)120   idx1 = RandomInteger[{1, Length[p1Collection]}];121   (*计算p2集合中的有效索引范围*)minIdx2 = Max[1, idx1 - 1];122   maxIdx2 = Min[Length[p2Collection], idx1 + 6];123   (*在有效范围内随机选择p2索引*)idx2 = RandomInteger[{minIdx2, maxIdx2}];124   (*获取点*)randomP1 = p1Collection[[idx1]];125   randomP2 = p2Collection[[idx2]];126   (*计算位置差*)posDiff = Abs[idx1 - idx2];127   (*创建颜色-位置差越小越蓝，越大越红*)hueValue = Rescale[posDiff, {0, 10}, {0.6, 0}];128   (*lineColor=Hue[hueValue];*)lineColor = Blue;129   (*绘制线段和端点*){Thickness[0.002], Opacity[0.7], lineColor, 130    Line[{randomP1, randomP2}]}, {30}]; (*循环30次*)131 132crOneRandomP2Lines = 133  Table[(*随机选择p1集合中的索引*)134   idx1 = RandomInteger[{1, Length[p2Collection]}];135   (*计算p2集合中的有效索引范围*)minIdx2 = Max[1, idx1 - 1];136   maxIdx2 = Min[Length[p2Collection], idx1 + 6];137   (*在有效范围内随机选择p2索引*)idx2 = RandomInteger[{minIdx2, maxIdx2}];138   (*获取点*)randomP1 = p2Collection[[idx1]];139   randomP2 = p2Collection[[idx2]];140   (*计算位置差*)posDiff = Abs[idx1 - idx2];141   (*创建颜色-位置差越小越蓝，越大越红*)hueValue = Rescale[posDiff, {0, 10}, {0.6, 0}];142   lineColor = Blue;143   (*绘制线段和端点*){Thickness[0.002], Opacity[0.7], lineColor, 144    Line[{randomP1, randomP2}]}, {30}]; (*循环30次*)145 146crOneRandomP3P4Lines = 147  Table[(*随机选择p1集合中的索引*)148   idx1 = RandomInteger[{1, Length[p3Collection]}];149   (*计算p2集合中的有效索引范围*)minIdx2 = Max[1, idx1 - 8];150   maxIdx2 = Min[Length[p4Collection], idx1];151   (*在有效范围内随机选择p2索引*)idx2 = RandomInteger[{minIdx2, maxIdx2}];152   (*获取点*)randomP1 = p3Collection[[idx1]];153   randomP2 = p4Collection[[idx2]];154   (*计算位置差*)posDiff = Abs[idx1 - idx2];155   (*创建颜色-位置差越小越蓝，越大越红*)hueValue = Rescale[posDiff, {0, 10}, {0.6, 0}];156   lineColor = Blue;157   (*绘制线段和端点*){Thickness[0.002], Opacity[0.7], lineColor, 158    Line[{randomP1, randomP2}]}, {30}]; (*循环30次*)159 160p1Collection = {};161p2Collection = {};162p3Collection = {};163p4Collection = {};164offset = Pi/32;165crTwoOffsets = {0 + offset, Pi/16 + offset, 166   Pi/28 + offset, -Pi/8 + offset};167crTwoTs = Range[0, 2 Pi, step];168crTwoConnectors = Table[random = RandomReal[{1, 15}]*0.02;169   p1 = {2 Cos[t + crTwoOffsets[[1]] + random], 170     1.5 Sin[t + crTwoOffsets[[1]] + random], 171     0.2 Sin[2 (t + crTwoOffsets[[1]]) + 3 Pi/2] + 3};172   p2 = {4 Cos[t + crTwoOffsets[[2]]], 4.2 Sin[t + crTwoOffsets[[2]]],173      0.6 Sin[2 (t + crTwoOffsets[[2]]) + 3 Pi/2] + 3.2};174   p3 = {4.2 Cos[t + crTwoOffsets[[3]]], 175     4.4 Sin[t + crTwoOffsets[[3]]], 176     0.6 Sin[2 (t + crTwoOffsets[[3]]) + 3 Pi/2] + 2.5};177   p4 = {3.4 Cos[t + crTwoOffsets[[4]] + random], 178     3.8 Sin[t + crTwoOffsets[[4]] + random], 0};179   AppendTo[p1Collection, p1];180   AppendTo[p2Collection, p2];181   AppendTo[p3Collection, p3];182   AppendTo[p4Collection, p4];183   pts = {p1, p2, p3, p4};184   newPts = 185    Join[{2.1 pts[[1]] - pts[[2]]},(*p0=2*p1-p2*)186     pts, {2.2 pts[[-1]] - pts[[-2]]}];187   crPts = 188    Flatten[Table[189      Table[catmullRom3D[newPts[[i ;; i + 3]], s], {s, 0, 1, 190        1/30}], {i, 1, Length[newPts] - 3}], 1];191   {Thickness[0.002], Opacity[0.7], Blue, Line[crPts], Transparent, 192    PointSize[0.001], Point[pts]}, {t, crTwoTs}];193 194crTwoConnectors1 = Table[random = RandomReal[{1, 15}]*0.02;195   p1 = {2 Cos[t + crTwoOffsets[[1]] + random], 196     1.5 Sin[t + crTwoOffsets[[1]] + random], 197     0.2 Sin[2 (t + crTwoOffsets[[1]]) + 3 Pi/2] + 3};198   p2 = {4 Cos[t + crTwoOffsets[[2]]], 4.2 Sin[t + crTwoOffsets[[2]]],199      0.6 Sin[2 (t + crTwoOffsets[[2]]) + 3 Pi/2] + 3.2};200   p3 = {4.2 Cos[t + crTwoOffsets[[3]] + step], 201     4.4 Sin[t + crTwoOffsets[[3]] + step], 202     0.6 Sin[2 (t + crTwoOffsets[[3]] + step) + 3 Pi/2] + 2.5};203   p4 = {3.4 Cos[t + crTwoOffsets[[4]] + random + step], 204     3.8 Sin[t + crTwoOffsets[[4]] + random + step], 0};205   pts = {p1, p2, p3, p4};206   crPts = 207    Flatten[Table[208      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i, 209       1, Length[pts] - 3}], 1];210   {Thickness[0.002], Opacity[0.7], Blue, Line[crPts], Transparent, 211    PointSize[0.001], Point[pts]}, {t, crTwoTs}];212 213 214crTwoConnectors2 = Table[random = RandomReal[{1, 15}]*0.02;215   p1 = {2 Cos[t + crTwoOffsets[[1]] + random], 216     1.5 Sin[t + crTwoOffsets[[1]] + random], 217     0.2 Sin[2 (t + crTwoOffsets[[1]]) + 3 Pi/2] + 3};218   p2 = {4 Cos[t + crTwoOffsets[[2]] + step], 219     4.2 Sin[t + crTwoOffsets[[2]] + step], 220     0.6 Sin[2 (t + crTwoOffsets[[2]] + step) + 3 Pi/2] + 3.2};221   p3 = {4.2 Cos[t + crTwoOffsets[[3]]], 222     4.4 Sin[t + crTwoOffsets[[3]]], 223     0.6 Sin[2 (t + crTwoOffsets[[3]]) + 3 Pi/2] + 2.5};224   p4 = {3.4 Cos[t + crTwoOffsets[[4]] + random - step], 225     3.8 Sin[t + crTwoOffsets[[4]] + random - step], 0};226   pts = {p1, p2, p3, p4};227   crPts = 228    Flatten[Table[229      Table[catmullRom3D[pts[[i ;; i + 3]], s], {s, 0, 1, 1/30}], {i, 230       1, Length[pts] - 3}], 1];231   {Thickness[0.002], Opacity[0.7], Blue, Line[crPts], Transparent, 232    PointSize[0.001], Point[pts]}, {t, crTwoTs}];233 234(*生成30条随机连接线，确保位置差不超过30*)235crTwoRandomP1P2Lines = 236  Table[(*随机选择p1集合中的索引*)237   idx1 = RandomInteger[{1, Length[p1Collection]}];238   (*计算p2集合中的有效索引范围*)minIdx2 = Max[1, idx1 - 6];239   maxIdx2 = Min[Length[p2Collection], idx1 + 1];240   (*在有效范围内随机选择p2索引*)idx2 = RandomInteger[{minIdx2, maxIdx2}];241   (*获取点*)randomP1 = p1Collection[[idx1]];242   randomP2 = p2Collection[[idx2]];243   (*计算位置差*)posDiff = Abs[idx1 - idx2];244   (*创建颜色-位置差越小越蓝，越大越红*)hueValue = Rescale[posDiff, {0, 10}, {0.6, 0}];245   lineColor = Blue;246   (*绘制线段和端点*){Thickness[0.002], Opacity[0.7], lineColor, 247    Line[{randomP1, randomP2}]}, {30}]; (*循环30次*)248 249 250 251crTwoRandomP2Lines = 252  Table[(*随机选择p1集合中的索引*)253   idx1 = RandomInteger[{1, Length[p2Collection]}];254   (*计算p2集合中的有效索引范围*)minIdx2 = Max[1, idx1 - 6];255   maxIdx2 = Min[Length[p2Collection], idx1 + 1];256   (*在有效范围内随机选择p2索引*)idx2 = RandomInteger[{minIdx2, maxIdx2}];257   (*获取点*)randomP1 = p2Collection[[idx1]];258   randomP2 = p2Collection[[idx2]];259   (*计算位置差*)posDiff = Abs[idx1 - idx2];260   (*创建颜色-位置差越小越蓝，越大越红*)hueValue = Rescale[posDiff, {0, 10}, {0.6, 0}];261   lineColor = Blue;262   (*绘制线段和端点*){Thickness[0.002], Opacity[0.7], lineColor, 263    Line[{randomP1, randomP2}]}, {30}]; (*循环30次*)264 265crTwoRandomP3P4Lines = 266  Table[(*随机选择p1集合中的索引*)267   idx1 = RandomInteger[{1, Length[p3Collection]}];268   (*计算p2集合中的有效索引范围*)minIdx2 = Max[1, idx1];269   maxIdx2 = Min[Length[p4Collection], idx1 + 8];270   (*在有效范围内随机选择p2索引*)idx2 = RandomInteger[{minIdx2, maxIdx2}];271   (*获取点*)randomP1 = p3Collection[[idx1]];272   randomP2 = p4Collection[[idx2]];273   (*计算位置差*)posDiff = Abs[idx1 - idx2];274   (*创建颜色-位置差越小越蓝，越大越红*)hueValue = Rescale[posDiff, {0, 10}, {0.6, 0}];275   lineColor = Blue;276   (*绘制线段和端点*){Thickness[0.002], Opacity[0.7], lineColor, 277    Line[{randomP1, randomP2}]}, {30}]; (*循环30次*)278 279 280 281png = Show[Graphics3D[crOneConnectors], Graphics3D[crOneConnectors1], 282  Graphics3D[crOneConnectors2], Graphics3D[crTwoConnectors], 283  Graphics3D[crTwoConnectors1], Graphics3D[crTwoConnectors2], 284  Graphics3D[crOneRandomP1P2Lines], Graphics3D[crOneRandomP2Lines], 285  Graphics3D[crOneRandomP3P4Lines], Graphics3D[crTwoRandomP1P2Lines], 286  Graphics3D[crTwoRandomP2Lines], Graphics3D[crTwoRandomP3P4Lines], 287  curve12,(*curve21,curve22,*)curve31, axes, Axes -> True, 288  AxesLabel -> {"X", "Y", "Z"}, 289  PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, BoxRatios -> {1, 1, 1}, 290  ImageSize -> Large, PlotLabel -> "Bird's Nest"]

最终效果：

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

A 3D graph projected onto the xy-plane