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从立方体到四面体

从立方体到四面体
几何画板及绘图内容加载中，请稍等……
2022-01-20 19:42
解析几何立体几何微积分几何作图
①本范例演示了从立方体通过连续变换生成四面体的过程；②涉及的核心数学概念包括立体几何、连续性、微积分、坐标几何；③作图要点提炼自构造摘要，使用了点、立方体、球体、相交、多边形、线段、旋转等GeoGebra命令与对象；④适合高中及以上学生及数学爱好者学习空间几何变换与微积分应用。

GeoGebra 命令及参数

  1. Point(Circle[A, Distance[B, A], Segment[B, A]]) → I
  2. Cube(B, A, I) → a, J, faceIJNM, faceKLMN, edgeBJ, edgeAB, edgeAI, edgeIJ, edgeBK, edgeJN, edgeKN, edgeKL, K, edgeAL, edgeLM, edgeIM, edgeMN, L, M, N, faceABJI, faceBJNK, faceABKL, faceAIML
  3. Sphere(G, s) → b
  4. Sphere(F, s) → c
  5. Sphere(H, s) → d
  6. Sphere(E, s) → e
  7. Sphere(I, s) → f
  8. Sphere(C, s) → g
  9. Sphere(B, s) → h
  10. Sphere(A, s) → i
  11. Intersect(edgeKL, e) → O, O_1
  12. Intersect(edgeKL, c) → P_1, P
  13. Intersect(edgeAB, h) → Q_1, Q
  14. Intersect(edgeAB, i) → R, R_1
  15. Intersect(edgeLM, d) → S, S_1
  16. Intersect(edgeMN, b) → T, T_1
  17. Intersect(edgeMN, d) → U_1, U
  18. Intersect(edgeIM, d) → V_1, V
  19. Intersect(edgeKN, b) → W_1, W
  20. Intersect(edgeJN, b) → Z, Z_1
  21. Intersect(edgeAL, e) → A_2, A_1
  22. Intersect(edgeLM, e) → B_2, B_1
  23. Intersect(edgeIM, f) → C_1, C_2
  24. Intersect(edgeIJ, f) → D_2, D_1
  25. Intersect(edgeIJ, g) → E_1, E_2
  26. Intersect(edgeJN, g) → F_2, F_1
  27. Intersect(edgeBJ, g) → G_2, G_1
  28. Intersect(edgeBJ, h) → H_1, H_2
  29. Intersect(edgeBK, h) → I_1, I_2
  30. Intersect(edgeBK, c) → J_2, J_1
  31. Intersect(edgeKN, c) → K_1, K_2
  32. Polygon(K_1, J_1, P) → t1, p, k_1, j_1
  33. Polygon(B_1, O, A_1) → t2, a_1, b_1, o
  34. Polygon(S, V, U) → t3, u, s_1, v
  35. Polygon(T, Z, W) → t4, w, t, z_1
  36. Intersect(edgeAL, i) → L_1, L_2
  37. Intersect(edgeAI, f) → M_1, M_2
  38. Intersect(edgeAI, i) → N_2, N_1
  39. Polygon(L_1, N_1, R) → t5, r, l_1, n_1
  40. Polygon(D_1, M_1, C_1) → t6, c_1, d_1, m_1
  41. Polygon(F_1, G_1, E_1) → t7, e_1, f_1, g_1
  42. Polygon(I_1, H_1, Q) → t8, q, i_1, h_1
  43. Segment(K_1, W) → j
  44. Segment(T, U) → k
  45. Segment(S, B_1) → l
  46. Segment(J_1, I_1) → m
  47. Segment(P, O) → n
  48. Segment(Z, F_1) → p_1
  49. Segment(A_1, L_1) → q_1
  50. Segment(R, Q) → r_1
  51. Segment(G_1, H_1) → t_1
  52. Segment(D_1, E_1) → f_2
  53. Segment(V, C_1) → g_2
  54. Segment(N_1, M_1) → h_2
  55. Segment(T, Z) → i_2
  56. Segment(F_1, E_1) → j_2
  57. Segment(C_1, D_1) → k_2
  58. Segment(V, U) → l_2
  59. Intersect(l_2, i_2) → O_2
  60. Intersect(l_2, k_2) → P_2
  61. Intersect(j_2, k_2) → Q_2
  62. Intersect(j_2, i_2) → R_2
  63. Polygon(O_2, P_2, Q_2, R_2) → q1, o_2, p_2, q_2, r_2
  64. Rotate(q1, 90°, zAxis) → q1'
  65. Rotate(q1, 180°, zAxis) → q1'_{1}
  66. Rotate(q1, 270°, zAxis) → q1'_{2}
  67. Rotate(O_2, 90°, zAxis) → S_2
  68. Rotate(P_2, 90°, zAxis) → T_2
  69. Rotate(Q_2, 90°, zAxis) → U_2
  70. Rotate(R_2, 90°, zAxis) → V_2
  71. Rotate(O_2, 180°, zAxis) → W_2
  72. Rotate(P_2, 180°, zAxis) → Z_2
  73. Rotate(Q_2, 180°, zAxis) → A_3
  74. Rotate(R_2, 180°, zAxis) → B_3
  75. Rotate(O_2, 270°, zAxis) → C_3
  76. Rotate(P_2, 270°, zAxis) → D_3
  77. Rotate(Q_2, 270°, zAxis) → E_3
  78. Rotate(R_2, 270°, zAxis) → F_3
  79. Rotate(q1, 90°, yAxis) → q1'_{3}
  80. Rotate(O_2, 90°, yAxis) → G_3
  81. Rotate(P_2, 90°, yAxis) → H_3
  82. Rotate(Q_2, 90°, yAxis) → I_3
  83. Rotate(R_2, 90°, yAxis) → J_3
  84. Rotate(q1, -90°, yAxis) → q1'_{4}
  85. Rotate(O_2, -90°, yAxis) → K_3
  86. Rotate(P_2, -90°, yAxis) → L_3
  87. Rotate(Q_2, -90°, yAxis) → M_3
  88. Rotate(R_2, -90°, yAxis) → N_3
  89. Polygon(R_2, T_2, S_2, N_3, M_3, O_2) → poly1, m_2, t_2, s_2, n_3, m_3, n_2
  90. Polygon(N_3, K_3, W_2, Z_2, V_2, S_2) → poly2, a_2, k_3, w_2, z_2, v_2, b_2
  91. Polygon(W_2, B_3, D_3, C_3, L_3, K_3) → poly3, c_2, b_3, d_3, c_3, l_3, d_2
  92. Polygon(C_3, F_3, P_2, O_2, M_3, L_3) → poly4, e_2, f_3, g_3, h_3, i_3, j_3
  93. Polygon(U_2, J_3, I_3, A_3, Z_2, V_2) → poly5, u_2, p_3, q_3, a_3, r_3, s_3
  94. Polygon(A_3, I_3, H_3, E_3, D_3, B_3) → poly6, t_3, e_3, f_4, g_4, h_4, i_4
  95. Polygon(E_3, H_3, G_3, Q_2, P_2, F_3) → poly7, j_4, k_4, l_4, m_4, n_4, p_4
  96. Polygon(R_2, Q_2, G_3, J_3, U_2, T_2) → poly8, q_4, r_4, s_4, t_4, a_4, b_4
  97. Polygon(S, U, T, W, K_1, P, O, B_1) → poly9, s_5, u_1, t_5, w_1, c_4, p_5, o_1, d_4
  98. Rotate(poly9, 90°, xAxis) → poly9'
  99. Rotate(poly9, 180°, xAxis) → poly9'_{1}
  100. Rotate(poly9, 270°, xAxis) → poly9'_{2}
  101. Rotate(K_1, -90°, yAxis) → O_3
  102. Rotate(J_1, -90°, yAxis) → P_3
  103. Segment(O_3, P_3) → p'
  104. Rotate(poly9'_{2}, 90°, zAxis) → poly10
  105. Rotate(poly9'_{2}, -90°, zAxis) → poly11

AI 绘图提示语参考

从一个立方体开始，逐步将其变形为一个四面体。首先创建一个立方体，然后通过旋转和截取的方式，逐步改变立方体的形状，最终形成一个四面体。注意观察每个步骤中关键点和面的变化。
以上提示语为根据原始绘图命令整理而来，仅供参考。