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伊斯兰几何图案的对称分割与构造

伊斯兰几何图案的对称分割与构造
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2019-12-31 18:01
多边形角变换几何作图
① 本范例演示了如何通过对称、旋转和镜像等几何变换，将基础多边形分割并组合成具有伊斯兰风格的复杂几何图案。② 核心数学概念包括对称性、中位线、角平分线、旋转与镜像变换，以及勾股定理在构造中的潜在应用。③ 作图要点：使用GeoGebra的Polygon、Point、Segment、Circle、Ray、Line等基础对象，结合Rotate、Mirror、OrthogonalLine、AngularBisector等变换与辅助线命令，并通过Intersect求交点，利用Textfield和ConstructionStep实现交互式步骤展示。④ 适合中学至大学阶段的几何教学，尤其适用于对称性、变换几何和图案设计课程，也可用于艺术与数学融合的跨学科项目。

GeoGebra 命令及参数

  1. Intersect(xAxis, yAxis) → A
  2. ConstructionStep() → stap
  3. Polygon(A, B, 4) → veelhoek1, f, g, h, i, C, D
  4. Point(i) → E
  5. Segment(A, E) → j
  6. Rotate(F, 90°, G) → F'
  7. Rotate(F, (-(90°)), G) → F'_1
  8. Mirror(F, G) → F'_2
  9. Segment(F, F'_1) → k
  10. Mirror(D, k) → D'
  11. Rotate(D', 90°, G) → D''
  12. Rotate(D', (-(90°)), G) → D''_1
  13. Mirror(D', G) → D''_2
  14. Segment(F, D') → l
  15. Segment(F'_1, D''_1) → m
  16. Segment(F'_2, D''_2) → n
  17. Segment(F', D'') → p
  18. Segment(F'_1, F'_2) → q
  19. Segment(F'_2, F') → r
  20. Segment(F', F) → s
  21. OrthogonalLine(D''_1, r) → t
  22. Intersect(t, r) → H
  23. Segment(D''_1, H) → d
  24. Segment(D''_2, D''_1) → a
  25. Textfield(F) → Invulvak1
  26. OrthogonalLine(D'', k) → b
  27. Intersect(b, k) → I
  28. Segment(I, D'') → c
  29. OrthogonalLine(D''_2, s) → e
  30. Intersect(e, s) → J
  31. Segment(D''_2, J) → f_1
  32. OrthogonalLine(D', q) → g_1
  33. Intersect(g_1, q) → K
  34. Segment(D', K) → h_1
  35. Segment(F, F'_1) → i_1
  36. Segment(F'_1, F'_2) → j_1
  37. Segment(F'_2, F') → k_1
  38. Segment(F', F) → l_1
  39. Segment(D''_1, H) → m_1
  40. Intersect(b, i) → L
  41. Segment(L, I) → n_1
  42. Intersect(e, f) → M
  43. Segment(J, M) → p_1
  44. Intersect(g_1, h) → N
  45. Segment(K, N) → q_1
  46. Intersect(t, g) → O
  47. Segment(H, O) → r_1
  48. Segment(L, F'_1) → s_1
  49. Circle(H, D''_1) → c_1
  50. Intersect(r, k_1) → P
  51. Intersect(r, k_1) → Q
  52. Intersect(r, k_1) → R
  53. Intersect(r, k_1) → S
  54. Intersect(c_1, k_1, 2) → T
  55. Circle(F', T) → d_1
  56. Intersect(d_1, p, 1) → U
  57. Segment(J, U) → t_1
  58. Intersect(d_1, f, 1) → V
  59. Segment(J, V) → a_1
  60. Intersect(c_1, n, 1) → W
  61. AngularBisector(U, D''_2, D''_1) → b_1
  62. AngularBisector(D''_1, H, T) → e_1
  63. Intersect(b_1, e_1) → Z
  64. Segment(D''_1, Z) → f_2
  65. Segment(Z, U) → g_2
  66. Segment(Z, T) → h_2
  67. Circle(K, N) → c_2
  68. Intersect(q, j_1) → A_1
  69. Intersect(c_2, j_1, 2) → B_1
  70. Circle(F'_2, B_1) → d_2
  71. Intersect(d_2, g, 2) → C_1
  72. AngularBisector(B_1, K, N) → i_2
  73. AngularBisector(C_1, F'_2, B_1) → j_2
  74. Intersect(i_2, j_2) → D_1
  75. Segment(B_1, D_1) → k_2
  76. Segment(D_1, C_1) → l_2
  77. Segment(D_1, N) → m_2
  78. AngularBisector(B_1, F'_2, W) → n_2
  79. AngularBisector(D', K, B_1) → p_2
  80. Intersect(n_2, p_2) → E_1
  81. Segment(D', E_1) → q_2
  82. Segment(E_1, B_1) → r_2
  83. Segment(E_1, W) → s_2
  84. Circle(I, D'') → e_2
  85. Circle(D', D'') → t_2
  86. Intersect(k, i_1) → F_1
  87. Intersect(k, i_1) → G_1
  88. Intersect(k, i_1) → H_1
  89. Intersect(k, i_1) → I_1
  90. Intersect(e_2, i_1, 2) → J_1
  91. AngularBisector(D'', I, J_1) → a_2
  92. Intersect(t_2, m, 1) → K_1
  93. AngularBisector(K_1, D', D'') → b_2
  94. Intersect(a_2, b_2) → L_1
  95. Segment(D'', L_1) → f_3
  96. Segment(L_1, J_1) → g_3
  97. Segment(L_1, K_1) → h_3
  98. Segment(K_1, K) → i_3
  99. Circle(K, K_1) → c_3
  100. Circle(D'', D''_2) → d_3
  101. Intersect(d_3, l, 1) → M_1
  102. Circle(F, M_1) → e_3
  103. Intersect(s, l_1) → N_1
  104. Intersect(s, l_1) → O_1
  105. Intersect(e_3, l_1, 1) → P_1
  106. AngularBisector(M_1, F, P_1) → j_3
  107. AngularBisector(D''_2, J, P_1) → k_3
  108. Intersect(k_3, j_3) → Q_1
  109. Segment(M_1, Q_1) → l_3
  110. Segment(Q_1, P_1) → m_3
  111. Segment(Q_1, D''_2) → n_3
  112. Segment(M_1, I) → p_3
  113. Circle(F, M_1) → q_3
  114. Intersect(q_3, i, 1) → R_1
  115. Segment(R_1, I) → r_3
  116. Circle(A, M) → s_3
  117. Intersect(i, j) → S_1
  118. Intersect(i, j) → T_1
  119. Intersect(s_3, i, 1) → U_1
  120. AngularBisector(M, A, U_1) → t_3
  121. AngularBisector(P_1, J, M) → a_3
  122. Intersect(t_3, a_3) → V_1
  123. Segment(U_1, V_1) → b_3
  124. Segment(V_1, P_1) → f_4
  125. Segment(V_1, M) → g_4
  126. AngularBisector(J_1, I, L) → h_4
  127. Intersect(h_4, s_1) → W_1
  128. Segment(W_1, J_1) → i_4
  129. Circle(D, L) → c_4
  130. Intersect(c_4, h, 1) → Z_1
  131. Segment(W_1, Z_1) → j_4
  132. AngularBisector(N, K, F'_1) → k_4
  133. Mirror(Z_1, F'_1) → A_2
  134. Segment(A_2, K) → l_4
  135. AngularBisector(C, B, A) → m_4
  136. Point(m_4) → B_2
  137. Circle(B, B_2) → d_4
  138. Segment(B, B_2 + Vector[B, B_2]) → b_4
  139. Intersect(d_4, f, 1) → C_2
  140. Circle(C_2, B) → e_4
  141. Intersect(e_4, f, 2) → D_2
  142. Ray(D_2, B_2) → n_4
  143. Intersect(g, n_4) → E_2
  144. Segment(D_2, E_2) → p_4
  145. Circle(E_2, ll) → q_4
  146. Intersect(q_4, g, 1) → F_2
  147. Circle(F_2, ll) → r_4
  148. Intersect(r_4, g, 2) → G_2
  149. Segment(D_2, G_2) → s_4
  150. Segment(H_2, F') → a_4
  151. Line(C, s_4) → t_4
  152. Polygon(F, F'_1, F'_2, LL) → vierhoek1, f_5, f'_1, f'_2, g_5

AI 绘图提示语参考

先画一个正方形，以其中心为旋转点，将一条边上的点绕中心旋转正负九十度并镜像，得到三个对称点。连接这些点形成内部星形多边形，再通过垂线和角平分线构造出多个交点，并连接这些交点形成对称的伊斯兰风格几何图案。整体由正方形外框和内部对称分割的线段组成，强调旋转与镜像对称。
以上提示语为根据原始绘图命令整理而来，仅供参考。