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正非边形的奇妙构造

正非边形的奇妙构造
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2019-05-14 00:30
多边形向量几何作图
本示例展示了正非边形的动态构造，通过旋转、平移等变换方法，探索正非边形的性质。核心数学概念包括多边形、圆、向量和变换。通过GeoGebra命令如Polygon、Rotate、Translate等，可以动态调整非边形的边长和角度。适合高中几何学习者和数学爱好者。

GeoGebra 命令及参数

  1. Polygon(A, B, 9) → poly1, f, C, D, E, F, G, H, I, g, h, i, j, k, l, m, n
  2. Line(F, D) → p
  3. Line(F, B) → q
  4. Line(D, B) → r
  5. Segment(F, E) → s
  6. Circle(F, E) → c
  7. Intersect(c, q, 2) → J
  8. Line(E, I) → t
  9. Segment(F, G) → a
  10. Segment(G, H) → b
  11. Segment(H, I) → d
  12. Segment(I, A) → e
  13. Segment(A, B) → f_1
  14. Segment(B, C) → g_1
  15. Segment(C, D) → h_1
  16. Segment(D, E) → i_1
  17. Rotate(B, If[0 < o ≤ 1, ((o * 2) * π / 9), (2 * π / 9)], Center[Circle[D, E, F]]) → B'
  18. Rotate(C, If[0 < o ≤ 1, ((o * 2) * π / 9), (2 * π / 9)], Center[Circle[D, E, F]]) → C'
  19. Segment(B', C') → j_1
  20. Rotate(C, If[0 < o ≤ 1, ((o * 2) * π / 9), (2 * π / 9)], Center[Circle[D, E, F]]) → C'_1
  21. Rotate(D, If[0 < o ≤ 1, ((o * 2) * π / 9), (2 * π / 9)], Center[Circle[D, E, F]]) → D'
  22. Segment(C'_1, D') → k_1
  23. Rotate(D, If[0 < o ≤ 1, ((o * 2) * π / 9), (2 * π / 9)], Center[Circle[D, E, F]]) → D'_1
  24. Rotate(E, If[0 < o ≤ 1, ((o * 2) * π / 9), (2 * π / 9)], Center[Circle[D, E, F]]) → E'
  25. Segment(D'_1, E') → l_1
  26. Rotate(F, If[0 < o ≤ 1, ((o * 2) * π / 9), (2 * π / 9)], Center[Circle[D, E, F]]) → F'
  27. Rotate(E, If[0 < o ≤ 1, ((o * 2) * π / 9), (2 * π / 9)], Center[Circle[D, E, F]]) → E'_1
  28. Segment(F', E'_1) → s'
  29. Rotate(F, If[0 < o ≤ 1, ((o * 2) * π / 9), (2 * π / 9)], Center[Circle[D, E, F]]) → F'_1
  30. Rotate(G, If[0 < o ≤ 1, ((o * 2) * π / 9), (2 * π / 9)], Center[Circle[D, E, F]]) → G'
  31. Segment(F'_1, G') → a'
  32. Rotate(G, If[0 < o ≤ 1, ((o * 2) * π / 9), (2 * π / 9)], Center[Circle[D, E, F]]) → G'_1
  33. Rotate(H, If[0 < o ≤ 1, ((o * 2) * π / 9), (2 * π / 9)], Center[Circle[D, E, F]]) → H'
  34. Segment(G'_1, H') → b'
  35. Rotate(H, If[0 < o ≤ 1, ((o * 2) * π / 9), (2 * π / 9)], Center[Circle[D, E, F]]) → H'_1
  36. Rotate(I, If[0 < o ≤ 1, ((o * 2) * π / 9), (2 * π / 9)], Center[Circle[D, E, F]]) → I'
  37. Segment(H'_1, I') → d'
  38. Rotate(I, If[0 < o ≤ 1, ((o * 2) * π / 9), (2 * π / 9)], Center[Circle[D, E, F]]) → I'_1
  39. Rotate(A, If[0 < o ≤ 1, ((o * 2) * π / 9), (2 * π / 9)], Center[Circle[D, E, F]]) → A'
  40. Segment(I'_1, A') → e'
  41. Rotate(A, If[0 < o ≤ 1, ((o * 2) * π / 9), (2 * π / 9)], Center[Circle[D, E, F]]) → A'_1
  42. Rotate(B, If[0 < o ≤ 1, ((o * 2) * π / 9), (2 * π / 9)], Center[Circle[D, E, F]]) → B'_1
  43. Segment(A'_1, B'_1) → n_1
  44. Rotate(poly1, If[1.5 < o ≤ 2.5, (((o - 1.5) * 2) * π / 9), (2 * π / 9)], Center[Circle[A, B, C]]) → poly1'
  45. Dilate(E'_1, If[3 < o ≤ 4, 4 - o, 0], J) → K
  46. Dilate(J, If[4 < o ≤ 5, 5 - o, 0], A'_1) → J'
  47. Dilate(C'_1, If[5 < o ≤ 6, 6 - o, 0], A'_1) → L
  48. Rotate(E', If[7 < o ≤ 8, ((-(o - 7)) * π / 3), ((-π)) / 3], E'_1) → E''
  49. Rotate(C'_1, If[8 < o ≤ 9, (((o - 8) * 2) * π / 9), (2 * π / 9)], A'_1) → N
  50. Rotate(L, If[8 < o ≤ 9, (((o - 8) * 2) * π / 9), (2 * π / 9)], A'_1) → L'
  51. Rotate(B, If[3 < o ≤ 5, ((o - 3) * π), (2 * π)], Center[Circle[A, B, C]]) → B'_2
  52. Center(Circle[A, B, C]) → O
  53. CircleArc(O, B, B'_2) → c_1
  54. Circle(A, B, C) → d_1
  55. Midpoint(C'_1, D'_1) → P
  56. Midpoint(D'_1, E'_1) → Q
  57. Dilate(C'_1, If[5 < o ≤ 6, 6 - o, 0], P) → M
  58. Segment(C'_1, M) → q_1
  59. Rotate(C'_1, If[6 < o ≤ 7, ((-(o - 6)) * π), (-π)], P) → R
  60. Rotate(M, If[6 < o ≤ 7, ((-(o - 6)) * π), (-π)], P) → M'
  61. Segment(R, M') → r_1
  62. Rotate(R, If[7 < o ≤ 8, (((-(o - 7)) * 7) * π / 9), (-7 * π / 9)], D'_1) → R'
  63. Rotate(M', If[7 < o ≤ 8, (((-(o - 7)) * 7) * π / 9), (-7 * π / 9)], D'_1) → M''
  64. Segment(R', M'') → t_1
  65. Rotate(R', If[8 < o ≤ 9, ((-(o - 8)) * π), (-π)], M'') → R''
  66. Rotate(M'', If[8 < o ≤ 9, ((-(o - 8)) * π), (-π)], M'') → M'''
  67. Segment(R'', M''') → m_1
  68. Line(O, M'') → p_1
  69. Intersect(d_1, p_1, 2) → S
  70. Midpoint(O, S) → T
  71. Dilate(O, If[10 < o ≤ 11, 11 - o, 0], T) → O'
  72. Segment(O, O') → s_1
  73. Dilate(T, If[11 < o ≤ 12, 12 - o, 0], S) → T'
  74. Segment(T, T') → a_1
  75. Rotate(O, If[12 < o ≤ 13, ((o - 12) * π), π], T) → O'_1
  76. Rotate(O', If[12 < o ≤ 13, ((o - 12) * π), π], T) → O''
  77. Segment(O'_1, O'') → b_1
  78. Dilate(T, If[13 < o ≤ 14, 14 - o, 0], P) → T'_1
  79. Dilate(O, If[13 < o ≤ 14, 14 - o, 0], P) → O'_2
  80. Segment(O, O'_2) → e_1
  81. Segment(T, T'_1) → f_2
  82. Line(G', C'_1) → g_2
  83. OrthogonalLine(F', g_2) → h_2
  84. Intersect(h_2, g_2) → U
  85. Angle(C'_1, U, F') → α
  86. Segment(G', U) → i_2
  87. Segment(F', U) → j_2
  88. Rotate(U, 30°, G') → U'
  89. Angle(U, G', U') → β
  90. Ray(G', U') → k_2
  91. Circle(F', (0.6 * i_2)) → d_2
  92. Intersect(d_2, a', 1) → V
  93. Segment(G', U') → l_2
  94. Dilate(G', If[15 < o ≤ 16, 16 - o, 0], D) → G''
  95. Segment(G', G'') → m_2
  96. Dilate(F', If[15 < o ≤ 16, 16 - o, 0], U) → F''
  97. Segment(F', F'') → n_2
  98. Rotate(V, If[16 < o ≤ 17, ((o - 16) * π / 6), π / 6], F') → V'
  99. CircleSector(F', V, V') → c_2
  100. Dilate(G', If[17 < o ≤ 18, 18 - o, 0], U') → G''_1
  101. Segment(G', G''_1) → p_2
  102. Dilate(F', If[19 < o ≤ 20, 20 - o, 0], G') → F''_1
  103. Vector(F', F''_1) → u
  104. Translate(c_2, u) → e_2
  105. Rotate(e_2, If[19 < o ≤ 20, (((o - 19) * 5) * π / 6), (5 * π / 6)], F''_1) → q_2
  106. Rotate(q_2, If[21 < o ≤ 22, ((-(o - 21)) * π / 6), ((-π)) / 6], G') → r_2
  107. Line(G', e_1) → s_2
  108. Dilate(G', If[23 < o ≤ 24, 24 - o, 0], P) → G''_2
  109. Vector(G', G''_2) → v
  110. Translate(r_2, v) → t_2
  111. Rotate(t_2, If[23 < o ≤ 24, ((o - 23) * π), π], G''_2) → c_3