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穹顶六号：基于旋转与镜像的几何穹顶构造

穹顶六号：基于旋转与镜像的几何穹顶构造
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2020-05-22 20:23
多边形角三角函数变换几何作图
该范例演示了一个由多边形、线段、圆弧及镜像变换组合而成的穹顶状几何图形，通过旋转和对称操作生成具有规律性的结构。核心数学概念包括中线、自然数序列、角度、三角函数以及直角三角形性质，体现了几何变换与三角关系的综合应用。作图要点：使用GeoGebra的Polygon、Intersect、Segment、Point、Midpoint、Rotate、CircleArc、Line、Plane、Mirror、Sequence等命令，构建了O、A、B、C、D等大量点对象，以及多边形vierhoek1和线段a、b、c、d等，通过旋转和镜像生成对称的圆弧与直线，最终形成穹顶轮廓。适合高中或大学低年级学生用于探索几何变换、三角函数与构造性作图的结合，也可作为数学建模或艺术几何设计的参考案例。

GeoGebra 命令及参数

  1. Polygon(A, B, C, D) → vierhoek1, a, b, c, d
  2. Intersect(d, xAxis) → E
  3. Intersect(a, yAxis) → F
  4. Segment(E, F) → f
  5. Intersect(b, xAxis) → G
  6. Segment(F, G) → g
  7. Point(c) → H
  8. Segment(G, H) → h
  9. Segment(H, E) → i
  10. Midpoint(f) → I
  11. Midpoint(g) → J
  12. Midpoint(h) → K
  13. Midpoint(i) → L
  14. Segment(D, B) → j
  15. Rotate(C, 90°, j) → C'
  16. CircleArc(O, C, C') → e
  17. Point(e) → N
  18. Point(e) → P
  19. Point(e) → Q
  20. Point(e) → R
  21. Line(K, zAxis) → k
  22. Intersect(k, e, 1) → S
  23. Line(U, b) → l
  24. Line(V, b) → m
  25. Line(W, b) → n
  26. Plane(G, H, T) → p
  27. Intersect(n, p) → Z
  28. Intersect(m, p) → A_1
  29. Intersect(l, p) → B_1
  30. Polygon(G, C, W, Z) → vierhoek2, g_1, c_1, w, z_1
  31. Polygon(Z, W, V, A_1) → vierhoek3, z_2, w_1, v, a_1
  32. Polygon(B_1, A_1, V, U) → vierhoek4, b_1, q, v_1, u
  33. Polygon(B_1, U, T) → driehoek1, t, r, u_1
  34. Plane(C, A, M) → p_1
  35. Mirror(vierhoek2, p_1) → vierhoek2'
  36. Mirror(vierhoek3, p_1) → vierhoek3'
  37. Mirror(vierhoek4, p_1) → vierhoek4'
  38. Mirror(driehoek1, p_1) → driehoek1'
  39. Polygon(G, Z, A_1, B_1, T, C_1) → veelhoek1, g_2, z_3, s, f_1, t_1, h_1
  40. Mirror(veelhoek1, p_1) → veelhoek1'
  41. Sequence(Rotate[l1, (nn * 90°), zAxis], nn, 1, 3) → l2
  42. Sequence(Rotate[l3, (nn * 90°), zAxis], nn, 1, 3) → l4
  43. Mirror(C_1, D_1) → E_1
  44. Rotate(C_1, 90°, zAxis) → F_1
  45. Line(C_1, E_1) → i_1
  46. Rotate(F_1, (-(90°)), i_1) → G_1
  47. CircleArc(D_1, F_1, G_1) → d_1
  48. Point(d_1) → I_1
  49. Point(d_1) → J_1
  50. Point(d_1) → K_1
  51. Point(d_1) → L_1
  52. Point(d_1) → M_1
  53. Mirror(N_1, p_1) → S_1
  54. Mirror(O_1, p_1) → T_1
  55. Mirror(P_1, p_1) → U_1
  56. Mirror(Q_1, p_1) → V_1
  57. Mirror(R_1, p_1) → W_1
  58. Polygon(C_1, F_1, R_1, W_1) → vierhoek5, j_1, k_1, r_1, l_1
  59. Polygon(W_1, R_1, Q_1, V_1) → vierhoek6, m_1, n_1, q_1, s_1
  60. Polygon(V_1, Q_1, P_1, U_1) → vierhoek7, e_1, f_2, h_2, i_2
  61. Polygon(U_1, P_1, O_1, T_1) → vierhoek8, j_2, k_2, o_1, l_2
  62. Polygon(T_1, O_1, N_1, S_1) → vierhoek9, m_2, n_2, p_2, q_2
  63. Polygon(S_1, N_1, H_1) → driehoek2, r_2, s_2, t_2
  64. Segment(C, W) → a_2
  65. Segment(W, V) → b_2
  66. Segment(V, U) → c_2
  67. Segment(U, T) → d_2
  68. Sequence(Rotate[l7, (nn * 90°), zAxis], nn, 1, 3) → l8
  69. Segment(C_1, W_1) → e_2
  70. Segment(W_1, V_1) → f_3
  71. Segment(V_1, U_1) → g_3
  72. Segment(U_1, T_1) → h_3
  73. Segment(T_1, S_1) → i_3
  74. Segment(S_1, H_1) → j_3
  75. Sequence(Rotate[l9, (nn * 90°), zAxis], nn, 1, 3) → l10
  76. Segment(G, Z) → k_3
  77. Segment(Z, A_1) → l_3
  78. Segment(A_1, B_1) → m_3
  79. Segment(B_1, T) → n_3
  80. Sequence(Rotate[l11, (nn * 90°), zAxis], nn, 1, 3) → l12
  81. Mirror(l11, p_1) → l11'
  82. Sequence(Rotate[l11', (nn * 90°), zAxis], nn, 1, 3) → l13
  83. Segment(F_1, H) → p_3
  84. Sequence(Segment[Rotate[F_1, (nn * 90°), zAxis], Rotate[H, (nn * 90°), zAxis]], nn, 1, 3) → l14
  85. Segment(C_1, F_1) → q_3
  86. Sequence(Segment[Rotate[C_1, (nn * 90°), zAxis], Rotate[F_1, (nn * 90°), zAxis]], nn, 1, 3) → l15
  87. Sequence(Rotate[l5, (nn * 90°), zAxis], nn, 1, 3) → l6
  88. Segment(A, C) → r_3