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二十面体与气球

二十面体与气球
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2020-06-23 15:48
立体几何微积分几何作图
①本范例演示了二十面体与气球的几何关系及表面变化；②涉及核心数学概念包括立体几何、积分计算、方差分析等；③构造过程中运用了Intersect（相交）、Point（点）、Sphere（球体）等GeoGebra命令；④适合高中及以上学生及数学爱好者用于理解空间几何与微积分应用。

GeoGebra 命令及参数

  1. Intersect(zAxis, xAxis) → A
  2. Point(zAxis) → B
  3. Sphere(A, B) → a
  4. IntersectPath(xOyPlane, a) → c
  5. Intersect(xAxis, c) → C, D
  6. Intersect(yAxis, c) → E, F
  7. Intersect(xAxis, zAxis) → H
  8. Rotate(C, 72°, A, xOyPlane) → C'
  9. Angle(C, A, C', xOyPlane) → α
  10. Rotate(C', 72°, A, xOyPlane) → C''
  11. Angle(C', A, C'', xOyPlane) → β
  12. Rotate(C'', 72°, A, xOyPlane) → C'''
  13. Angle(C'', A, C''', xOyPlane) → γ
  14. Rotate(C''', 72°, A, xOyPlane) → G
  15. Angle(C''', A, G, xOyPlane) → δ
  16. CircleArc(A, B, C''') → d
  17. CircleArc(A, B, G) → e
  18. CircleArc(A, B, C) → f
  19. CircleArc(A, B, C') → g
  20. CircleArc(A, B, C'') → h
  21. Intersect(zAxis, a) → I, J
  22. Rotate(C, 36°, A, xOyPlane) → C'_1
  23. Angle(C, A, C'_1, xOyPlane) → ε
  24. Rotate(C'_1, 72°, A, xOyPlane) → K
  25. Angle(C'_1, A, K, xOyPlane) → ζ
  26. Rotate(K, 72°, A, xOyPlane) → K'
  27. Angle(K, A, K', xOyPlane) → η
  28. Rotate(K', 72°, A, xOyPlane) → K''
  29. Angle(K', A, K'', xOyPlane) → θ
  30. Rotate(K'', 72°, A, xOyPlane) → K'''
  31. Angle(K'', A, K''', xOyPlane) → ι
  32. CircleArc(A, I, K'') → k
  33. Point(zAxis) → L
  34. CircleArc(L, K', I) → p
  35. CircleArc(L, K''', I) → q
  36. CircleArc(L, K, I) → r
  37. CircleArc(L, C'_1, I) → s
  38. Distance(A, B) → AB
  39. Sphere(B, i) → j
  40. IntersectConic(j, a) → t
  41. Intersect(f, t) → M, N, ,
  42. Intersect(g, t) → O, P, ,
  43. Intersect(h, t) → Q, R, ,
  44. Intersect(t, d) → S, T, ,
  45. Intersect(t, e) → U, V, ,
  46. Sphere(I, i) → l
  47. IntersectConic(l, a) → c_1
  48. Intersect(q, c_1) → W, Z, ,
  49. Intersect(c_1, s) → A_1, B_1, ,
  50. Intersect(c_1, k) → C_1, D_1, ,
  51. Intersect(c_1, r) → E_1, F_1, ,
  52. Intersect(p, c_1) → G_1, H_1, ,
  53. CircleArc(L, B, N) → d_1
  54. CircleArc(L, N, U) → e_1
  55. CircleArc(L, B, U) → f_1
  56. CircleArc(L, N, W) → g_1
  57. Point(g_1) → I_1
  58. CircleArc(L, U, I_1) → h_1
  59. CircleArc(L, S, U) → k_1
  60. CircleArc(L, N, P) → p_1
  61. CircleArc(L, B, P) → q_1
  62. CircleArc(L, S, B) → r_1
  63. CircleArc(L, B, R) → s_1
  64. CircleArc(L, R, S) → t_1
  65. CircleArc(L, R, P) → c_2
  66. CircleArc(L, P, F_1) → d_2
  67. CircleArc(L, R, F_1) → e_2
  68. CircleArc(H, R, G_1) → f_2
  69. CircleArc(H, S, G_1) → g_2
  70. CircleArc(H, F_1, G_1) → h_2
  71. Intersect(t, k_1, 2) → J_1
  72. CircleArc(H, J_1, C_1) → k_2
  73. CircleArc(H, U, C_1) → p_2
  74. CircleArc(H, C_1, I_1) → q_2
  75. CircleArc(H, C_1, G_1) → r_2
  76. CircleArc(H, I_1, B_1) → s_2
  77. CircleArc(H, I_1, I) → t_2
  78. CircleArc(H, C_1, I) → c_3
  79. CircleArc(H, I, B_1) → d_3
  80. CircleArc(H, G_1, I) → e_3
  81. CircleArc(H, B_1, F_1) → f_3
  82. CircleArc(H, F_1, I) → g_3
  83. CircleArc(H, B_1, P) → h_3
  84. CircleArc(H, B_1, N) → k_3
  85. Intersect(t_1, t, 1) → K_1
  86. Polygon(B, R, S) → poly1, s_3, b_1, r_3
  87. Intersect(t, k_1, 1) → L_1
  88. Polygon(B, S, U) → poly2, u, b_2, s_4
  89. Polygon(S, C_1, L_1) → poly3, l_1, s_5, c_4
  90. Polygon(S, R, G_1) → poly4, g_4, s_6, r_4
  91. Polygon(S, G_1, C_1) → poly5, c_5, s_7, g_5
  92. Polygon(P, K_1, B) → poly6, b_3, p_3, k_4
  93. Polygon(B, N, U) → poly7, u_1, b_4, n
  94. Polygon(U, I_1, C_1) → poly8, c_6, u_2, i_1
  95. Polygon(P, N, B) → poly9, b_5, p_4, n_1
  96. Polygon(I_1, B_1, N) → poly10, n_2, i_2, b_6
  97. Polygon(N, P, B_1) → poly11, b_7, n_3, p_5
  98. Polygon(B_1, F_1, P) → poly12, p_6, b_8, f_4
  99. Polygon(P, F_1, K_1) → poly13, k_5, p_7, f_5
  100. Polygon(F_1, G_1, R) → poly14, r_5, f_6, g_6
  101. Polygon(F_1, I, G_1) → poly15, g_7, f_7, i_3
  102. Intersect(c_1, h_3, 1) → M_1
  103. Polygon(B_1, F_1, I) → poly16, i_4, b_9, f_8
  104. Polygon(I_1, B_1, I) → poly17, i_5, i_6, b_{10}
  105. Polygon(G_1, I, C_1) → poly18, c_7, g_8, i_7
  106. Intersect(c_3, q_2, 1) → N_1
  107. Polygon(I_1, C_1, I) → poly19, i_8, i_9, c_8
  108. Polygon(L_1, N, I_1) → t1, j_1, m_1, n_4