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椭圆镶嵌：用仿射变换与圆锥曲线生成图案

椭圆镶嵌：用仿射变换与圆锥曲线生成图案
几何画板及绘图内容加载中，请稍等……
2023-11-22 10:44
多边形变换几何作图函数图像
本范例通过几何镶嵌的方式展示如何用椭圆形成连续的图案与层叠效果。核心概念包括圆锥曲线（Conic）中椭圆的构造、几何变换（Mirror 镜像、Dilate 放缩）、以及由多边形与线段/直线的交点、对称与中点等要素来精确定位曲线生成的关键点，从而实现图形的“拼接”与重复。作图要点：使用 Polygon/Segment/Line 形成基础几何结构；用 Intersect 计算曲线与边界相关的交点；用 Midpoint 确定对称轴或缩放中心；用 Mirror 完成镜像配对，再用 Dilate 将局部椭圆形态按比例扩展以生成镶嵌单元；最后通过 Conic 按指定五元组/几何条件创建椭圆，并以 Element 管理可视对象；界面对象还包含按钮按钮1与多组参数变量（如 a、b、c、d、e、f、g、h 等）以便交互调参。适合高中到大学基础几何、解析几何与几何变换的课堂演示，也适合数学竞赛训练中的几何构造与可视化探究。

GeoGebra 命令及参数

  1. Zip(Zip[(i + (0.7 * RandomUniform[0, w]), j + (0.7 * RandomUniform[0, h])), i, Lm], j, Lm) → PointsN
  2. Polygon(A, B, C, D) → q1, a, b, c, d
  3. Segment(B, D) → f
  4. Line(A, B) → g
  5. Line(B, C) → i
  6. Line(D, A) → j
  7. Intersect(h, g) → E
  8. Intersect(i, j) → F
  9. Segment(C, A) → k
  10. Intersect(f, k) → K
  11. Line(E, K) → m
  12. Line(F, K) → n
  13. Intersect(h, n) → G
  14. Intersect(j, m) → H
  15. Intersect(i, m) → I
  16. Intersect(g, n) → J
  17. Segment(G, H) → l
  18. Segment(G, I) → p
  19. Midpoint(l) → L
  20. Midpoint(p) → M
  21. Line(D, L) → q
  22. Line(C, M) → r
  23. Intersect(q, r) → N
  24. Mirror(G, N) → G'
  25. Conic(I, G, H, G', J) → e
  26. Dilate(e, 0.85, Center[e]) → e'
  27. Zip(Zip[Polygon[Element[PointsN, i, j], Element[PointsN, i, j + 1], Element[PointsN, i + 1, j + 1], Element[PointsN, i + 1, j]], i, Lm], j, Lm) → PolyN
  28. Zip(Zip[EllipseQ[Element[PointsN, i, j], Element[PointsN, i, j + 1], Element[PointsN, i + 1, j + 1], Element[PointsN, i + 1, j]], i, Ln], j, Ln) → EllQ
  29. Element(EllQ, 1, 1) → S1_1
  30. Element(EllQ, 2, 1) → S2_1
  31. Element(EllQ, 3, 1) → S3_1
  32. Element(EllQ, 4, 1) → S4_1
  33. Element(EllQ, 5, 1) → S5_1
  34. Element(EllQ, 6, 1) → S6_1
  35. Element(EllQ, 7, 1) → S7_1
  36. Element(EllQ, 8, 1) → S8_1
  37. Element(EllQ, 9, 1) → S9_1
  38. Element(EllQ, 1, 2) → S1_2
  39. Element(EllQ, 2, 2) → S2_2
  40. Element(EllQ, 3, 2) → S3_2
  41. Element(EllQ, 4, 2) → S4_2
  42. Element(EllQ, 5, 2) → S5_2
  43. Element(EllQ, 6, 2) → S6_2
  44. Element(EllQ, 7, 2) → S7_2
  45. Element(EllQ, 8, 2) → S8_2
  46. Element(EllQ, 9, 2) → S9_2
  47. Element(EllQ, 1, 3) → S1_3
  48. Element(EllQ, 2, 3) → S2_3
  49. Element(EllQ, 3, 3) → S3_3
  50. Element(EllQ, 4, 3) → S4_3
  51. Element(EllQ, 5, 3) → S5_3
  52. Element(EllQ, 6, 3) → S6_3
  53. Element(EllQ, 7, 3) → S7_3
  54. Element(EllQ, 8, 3) → S8_3
  55. Element(EllQ, 9, 3) → S9_3
  56. Element(EllQ, 1, 4) → S1_4
  57. Element(EllQ, 2, 4) → S2_4
  58. Element(EllQ, 3, 4) → S3_4
  59. Element(EllQ, 4, 4) → S4_4
  60. Element(EllQ, 5, 4) → S5_4
  61. Element(EllQ, 6, 4) → S6_4
  62. Element(EllQ, 7, 4) → S7_4
  63. Element(EllQ, 8, 4) → S8_4
  64. Element(EllQ, 9, 4) → S9_4
  65. Element(EllQ, 1, 5) → S1_5
  66. Element(EllQ, 2, 5) → S2_5
  67. Element(EllQ, 3, 5) → S3_5
  68. Element(EllQ, 4, 5) → S4_5
  69. Element(EllQ, 5, 5) → S5_5
  70. Element(EllQ, 6, 5) → S6_5
  71. Element(EllQ, 7, 5) → S7_5
  72. Element(EllQ, 8, 5) → S8_5
  73. Element(EllQ, 9, 5) → S9_5
  74. Element(EllQ, 1, 6) → S1_6
  75. Element(EllQ, 2, 6) → S2_6
  76. Element(EllQ, 3, 6) → S3_6
  77. Element(EllQ, 4, 6) → S4_6
  78. Element(EllQ, 5, 6) → S5_6
  79. Element(EllQ, 6, 6) → S6_6
  80. Element(EllQ, 7, 6) → S7_6
  81. Element(EllQ, 8, 6) → S8_6
  82. Element(EllQ, 9, 6) → S9_6
  83. Element(EllQ, 1, 7) → S1_7
  84. Element(EllQ, 2, 7) → S2_7
  85. Element(EllQ, 3, 7) → S3_7
  86. Element(EllQ, 4, 7) → S4_7
  87. Element(EllQ, 5, 7) → S5_7
  88. Element(EllQ, 6, 7) → S6_7
  89. Element(EllQ, 7, 7) → S7_7
  90. Element(EllQ, 8, 7) → S8_7
  91. Element(EllQ, 9, 7) → S9_7
  92. Element(EllQ, 1, 8) → S1_8
  93. Element(EllQ, 2, 8) → S2_8
  94. Element(EllQ, 3, 8) → S3_8
  95. Element(EllQ, 4, 8) → S4_8
  96. Element(EllQ, 5, 8) → S5_8
  97. Element(EllQ, 6, 8) → S6_8
  98. Element(EllQ, 7, 8) → S7_8
  99. Element(EllQ, 8, 8) → S8_8
  100. Element(EllQ, 9, 8) → S9_8
  101. Element(EllQ, 1, 9) → S1_9
  102. Element(EllQ, 2, 9) → S2_9
  103. Element(EllQ, 3, 9) → S3_9
  104. Element(EllQ, 4, 9) → S4_9
  105. Element(EllQ, 5, 9) → S5_9
  106. Element(EllQ, 6, 9) → S6_9
  107. Element(EllQ, 7, 9) → S7_9
  108. Element(EllQ, 8, 9) → S8_9
  109. Element(EllQ, 9, 9) → S9_9
  110. Zip(Zip[Centroid[Element[PolyN, i, j]], i, Lm], j, Lm) → CN

AI 绘图提示语参考

请绘制一个由多个椭圆组成的镶嵌图案，图案具有重复性和对称性。构造顺序：先绘制一个多边形框架，确定关键点；然后通过几何变换（如镜像和缩放）生成椭圆；最后将椭圆排列成重复的镶嵌图案。关键对象包括椭圆、点、线段和交点，椭圆由特定点确定，镶嵌通过缩放和镜像实现。
以上提示语为根据原始绘图命令整理而来，仅供参考。