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三维魔星动画

三维魔星动画
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2020-07-17 00:33
多边形向量立体几何几何作图
所属专题 Action Modular Origami
①这个范例演示了一个由纸折叠而成的三维魔星模型动画；②涉及的核心数学概念包括立体几何、变换、多边形和向量；③作图要点——从下方『构造摘要』提炼用到了 Rotate, Polygon, Intersect, Line, OrthogonalLine, Vector, Translate, Point, Dilate, Sequence 等GeoGebra命令和对象；④适合高中及以上学生学习和理解三维图形的构造与变换，也可用于数学教学演示。

GeoGebra 命令及参数

  1. Rotate(F, (-(45°)), F) → F'_1
  2. Rotate(C, (-(45°)), F) → C'_1
  3. Rotate(E, (-(45°)), F) → E'_1
  4. Rotate(D, (-(45°)), F) → D'_1
  5. Polygon(C'_1, D'_1, E'_1, F'_1) → poly1'_1, c'_1, d'_1, e'_1, f'_1
  6. Polygon(C, D, E, F) → poly1, c, d, e, f
  7. Intersect(e, d'_1) → W
  8. Intersect(d, d'_1) → V
  9. Intersect(xAxis, yAxis) → A
  10. Line(A, B) → a
  11. OrthogonalLine(F, a) → b
  12. Rotate(a, α / 2, A) → a'
  13. Intersect(a', b) → H
  14. Vector(F, H) → u
  15. Translate(V, u) → V'
  16. Translate(W, u) → W'
  17. Translate(F, u) → F'
  18. Translate(D, u) → D'
  19. Translate(C, u) → C'
  20. Point(a') → U
  21. Dilate(U, fac, A) → U'
  22. Translate(E, u) → E'
  23. Polygon(C', D', V', W', F') → poly1', g, h, a_1, b_1, c_1
  24. Vector(A, G) → v
  25. Translate(C', v) → C''
  26. Translate(D', v) → D''
  27. Translate(E', v) → E''
  28. Translate(F', v) → F''
  29. Polygon(C'', D'', E'', F'') → poly1'', c'', d'', e'', f''
  30. Rotate(C'', α, A) → C'''_1
  31. Rotate(D'', α, A) → D'''_1
  32. Rotate(E'', α, A) → E'''_1
  33. Rotate(F'', α, A) → F'''_1
  34. Polygon(C'''_1, D'''_1, E'''_1, F'''_1) → poly1'''_1, c''', d''', e''', f'''
  35. Rotate(C'', (2 * α), A) → C'''_2
  36. Rotate(D'', (2 * α), A) → D'''_2
  37. Rotate(E'', (2 * α), A) → E'''_2
  38. Rotate(F'', (2 * α), A) → F'''_2
  39. Polygon(C'''_2, D'''_2, E'''_2, F'''_2) → poly1'''_2, c'''_1, d'''_1, e'''_1, f'''_1
  40. Rotate(C'''_1, (2 * α), A) → I
  41. Rotate(D'''_1, (2 * α), A) → J
  42. Rotate(E'''_1, (2 * α), A) → K
  43. Rotate(F'''_1, (2 * α), A) → L
  44. Polygon(I, J, K, L) → poly2, i, j, k, l
  45. Rotate(C'', (4 * α), A) → C'''_3
  46. Rotate(D'', (4 * α), A) → D'''_3
  47. Rotate(E'', (4 * α), A) → E'''_3
  48. Rotate(F'', (4 * α), A) → F'''_3
  49. Polygon(C'''_3, D'''_3, E'''_3, F'''_3) → poly1'''_3, c'''_2, d'''_2, e'''_2, f'''_2
  50. Rotate(C'''_1, (4 * α), A) → M
  51. Rotate(D'''_1, (4 * α), A) → N
  52. Rotate(E'''_1, (4 * α), A) → O
  53. Rotate(F'''_1, (4 * α), A) → P
  54. Polygon(M, N, O, P) → poly3, m, n, o, p
  55. Rotate(C'', (6 * α), A) → C'''
  56. Rotate(D'', (6 * α), A) → D'''
  57. Rotate(E'', (6 * α), A) → E'''
  58. Rotate(F'', (6 * α), A) → F'''
  59. Polygon(C''', D''', E''', F''') → poly1''', c'''_3, d'''_3, e'''_3, f'''_3
  60. Rotate(C'''_1, (6 * α), A) → Q
  61. Rotate(D'''_1, (6 * α), A) → R
  62. Rotate(E'''_1, (6 * α), A) → S
  63. Rotate(F'''_1, (6 * α), A) → T
  64. Polygon(Q, R, S, T) → poly4, q, r, s, t
  65. Translate(V', v) → V''
  66. Translate(W', v) → W''
  67. Polygon(C'', D'', V'', W'', F'') → polyPiece, c''_1, d''_1, v'', w'', f''_1
  68. Sequence(Polygon[Rotate[C'', ((2 * n) * α), A], Rotate[D'', ((2 * n) * α), A], Rotate[V'', ((2 * n) * α), A], Rotate[W'', ((2 * n) * α), A], Rotate[F'', ((2 * n) * α), A]], n, 0, 3, 1) → ListPoly1
  69. Sequence(Polygon[Rotate[C'', α + ((2 * n) * α), A], Rotate[D'', α + ((2 * n) * α), A], Rotate[V'', α + ((2 * n) * α), A], Rotate[W'', α + ((2 * n) * α), A], Rotate[F'', α + ((2 * n) * α), A]], n, 0, 3, 1) → ListPoly2
  70. Vector(A, A') → w
  71. Translate(I, w) → I'
  72. Translate(J, w) → J'
  73. Polygon(J, J', I', I) → polySide, j_1, j', i', i_1
  74. Sequence(Rotate[polySide, α + ((2 * n) * α), A], n, 0, 3, 1) → ListPolySide1
  75. Sequence(Rotate[polySide, ((2 * n) * α), A], n, 0, 3, 1) → ListPolySide2
  76. Polygon(Q, R, V'', D'') → polyThin, q_1, r_1, v''_1, d''_2
  77. Polygon(F'', W'', V'', D'', C'') → poly6, f''_2, w''_1, v''_2, d''_3, c''_2
  78. Rotate(F'', (-α), A) → F'''_4
  79. Rotate(W'', (-α), A) → W'''
  80. Rotate(V'', (-α), A) → V'''
  81. Rotate(D'', (-α), A) → D'''_4
  82. Rotate(C'', (-α), A) → C'''_4
  83. Polygon(F'''_4, W''', V''', D'''_4, C'''_4) → poly6', f'''_4, w''', v''', d'''_4, c'''_4
  84. OrthogonalLine(F'''_1, c''_2) → f_1
  85. Intersect(f_1, v''_1) → Z
  86. Polygon(F'', W'', V'', Z, F'''_1) → poly7, f''_3, w''_2, v''_3, z_1, f'''_5
  87. Rotate(F'', α, A) → F'''_5
  88. Rotate(W'', α, A) → W'''_1
  89. Rotate(V'', α, A) → V'''_1
  90. Rotate(Z, α, A) → Z'
  91. Rotate(F'''_1, α, A) → A_1
  92. Polygon(F'''_5, W'''_1, V'''_1, Z', A_1) → polyFat, f'''_6, w'''_1, v'''_1, z', a_2
  93. Sequence(Polygon[Rotate[Q, ((2 * n) * α), A], Rotate[R, ((2 * n) * α), A], Rotate[V'', ((2 * n) * α), A], Rotate[D'', ((2 * n) * α), A]], n, 0, 3, 1) → ListPolyThin2
  94. Sequence(Polygon[Rotate[Q, α + ((2 * n) * α), A], Rotate[R, α + ((2 * n) * α), A], Rotate[V'', α + ((2 * n) * α), A], Rotate[D'', α + ((2 * n) * α), A]], n, 0, 3, 1) → ListPolyThin1
  95. Sequence(Polygon[Rotate[F'''_5, α + ((2 * n) * α), A], Rotate[W'''_1, α + ((2 * n) * α), A], Rotate[V'''_1, α + ((2 * n) * α), A], Rotate[Z', α + ((2 * n) * α), A], Rotate[A_1, α + ((2 * n) * α), A]], n, 0, 3, 1) → ListPolyFat1
  96. Sequence(Polygon[Rotate[F'''_5, ((2 * n) * α), A], Rotate[W'''_1, ((2 * n) * α), A], Rotate[V'''_1, ((2 * n) * α), A], Rotate[Z', ((2 * n) * α), A], Rotate[A_1, ((2 * n) * α), A]], n, 0, 3, 1) → ListPolyFat2
  97. Translate(ListPolyFat1, w) → ListPolyFat1'
  98. Translate(ListPolyThin2, w) → ListPolyThin2'
  99. Translate(ListPolyThin1, w) → ListPolyThin1'
  100. Translate(ListPolyFat2, w) → ListPolyFat2'

AI 绘图提示语参考

绘制一个3D魔星折纸动画，首先创建一个正六边形，然后围绕一个点旋转该六边形的各个顶点，生成一系列新的顶点。接着，通过这些顶点构造新的多边形，形成魔星的立体结构。最后，围绕中心点旋转这些多边形，生成动画效果。
以上提示语为根据原始绘图命令整理而来，仅供参考。