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GeoGebra 分形树与几何雪花：用递归与分段映射生成雪景

GeoGebra 分形树与几何雪花：用递归与分段映射生成雪景
几何画板及绘图内容加载中，请稍等……
2023-12-11 15:23
数列变换几何作图滑块动画
该范例通过 GeoGebra Snow 与分形树的组合动画，展示如何用递归式几何规则生成细节丰富的“雪景”图形，并观察其随迭代逐步展开的形态。核心数学概念包括分形几何（递归迭代）、分段函数/规则化生成、几何变换（平移与旋转等）、以及由角度与长度参数驱动的多边形边界增长。作图要点是：使用 RandomUniform 与 Random 产生随机或扰动参数，再借助 Translate 等变换将局部结构逐步定位与拼接；同时通过 initialAngle、time、angle、radius 以及一系列 posX/posY、size 相关变量管理每次迭代的方向、步长与缩放；对象 A、B、pic1 与按钮 button1/button2 用于交互触发与显示。对象列表中出现的 Ln 与多组 posX1~posX10、posY1~posY10、size1~size4 体现了迭代/层级的分段构造思路。适合中学竞赛社团、数学探究与可视化教学，也适用于想将分形与几何变换结合的 GeoGebra 学习者。

GeoGebra 命令及参数

  1. RandomUniform(2, 5) → size
  2. Random(0, (2 * pi)) → initialAngle
  3. Translate(Rotate[pic1, angle], Vector[(0, 0), (posX, posY)]) → S1
  4. RandomUniform(0.3, 1.2) → size1
  5. RandomUniform(0.3, 1.2) → size2
  6. RandomUniform(0.3, 1.2) → size3
  7. RandomUniform(0.3, 1.2) → size4
  8. RandomUniform(0.3, 1.2) → size5
  9. RandomUniform(0.3, 1.2) → size6
  10. RandomUniform(0.3, 1.2) → size7
  11. RandomUniform(0.3, 1.2) → size8
  12. RandomUniform(0.3, 1.2) → size9
  13. RandomUniform(0.3, 1.2) → size10
  14. RandomUniform(0, (2 * pi)) → initialAngle1
  15. Random(1, 3) → w1
  16. Random(1, 3) → w2
  17. Random(1, 3) → w3
  18. Random(1, 3) → w4
  19. Random(1, 3) → w5
  20. Random(1, 3) → w6
  21. Random(1, 3) → w7
  22. Random(1, 3) → w8
  23. Random(1, 3) → w9
  24. Random(1, 3) → w10
  25. RandomUniform(0, (2 * pi)) → initialAngle2
  26. RandomUniform(0, (2 * pi)) → initialAngle3
  27. RandomUniform(0, (2 * pi)) → initialAngle4
  28. RandomUniform(0, (2 * pi)) → initialAngle5
  29. RandomUniform(0, (2 * pi)) → initialAngle6
  30. RandomUniform(0, (2 * pi)) → initialAngle7
  31. RandomUniform(0, (2 * pi)) → initialAngle8
  32. RandomUniform(0, (2 * pi)) → initialAngle9
  33. RandomUniform(0, (2 * pi)) → initialAngle10
  34. RandomUniform(0.3, 1.2) → size11
  35. RandomUniform(0.3, 1.2) → size12
  36. RandomUniform(0.3, 1.2) → size13
  37. RandomUniform(0.3, 1.2) → size14
  38. RandomUniform(0.3, 1.2) → size15
  39. RandomUniform(0.3, 1.2) → size16
  40. RandomUniform(0.3, 1.2) → size17
  41. RandomUniform(0.3, 1.2) → size18
  42. RandomUniform(0.3, 1.2) → size19
  43. RandomUniform(0.3, 1.2) → size20
  44. Random(1, 3) → w11
  45. Random(1, 3) → w12
  46. Random(1, 3) → w13
  47. Random(1, 3) → w14
  48. Random(1, 3) → w15
  49. Random(1, 3) → w16
  50. Random(1, 3) → w17
  51. Random(1, 3) → w18
  52. Random(1, 3) → w19
  53. Random(1, 3) → w20
  54. RandomUniform(0, (2 * pi)) → initialAngle11
  55. RandomUniform(0, (2 * pi)) → initialAngle12
  56. RandomUniform(0, (2 * pi)) → initialAngle13
  57. RandomUniform(0, (2 * pi)) → initialAngle14
  58. RandomUniform(0, (2 * pi)) → initialAngle15
  59. RandomUniform(0, (2 * pi)) → initialAngle16
  60. RandomUniform(0, (2 * pi)) → initialAngle17
  61. RandomUniform(0, (2 * pi)) → initialAngle18
  62. RandomUniform(0, (2 * pi)) → initialAngle19
  63. RandomUniform(0, (2 * pi)) → initialAngle20
  64. RandomUniform(0, 10^(2)) → radius1
  65. RandomUniform(0, 10^(2)) → radius2
  66. RandomUniform(0, 10^(2)) → radius3
  67. RandomUniform(0, 10^(2)) → radius4
  68. RandomUniform(0, 10^(2)) → radius5
  69. RandomUniform(0, 10^(2)) → radius6
  70. RandomUniform(0, 10^(2)) → radius7
  71. RandomUniform(0, 10^(2)) → radius8
  72. RandomUniform(0, 10^(2)) → radius9
  73. RandomUniform(0, 10^(2)) → radius10
  74. RandomUniform(0, 10^(2)) → radius11
  75. RandomUniform(0, 10^(2)) → radius12
  76. RandomUniform(0, 10^(2)) → radius13
  77. RandomUniform(0, 10^(2)) → radius14
  78. RandomUniform(0, 10^(2)) → radius15
  79. RandomUniform(0, 10^(2)) → radius16
  80. RandomUniform(0, 10^(2)) → radius17
  81. RandomUniform(0, 10^(2)) → radius18
  82. RandomUniform(0, 10^(2)) → radius19
  83. RandomUniform(0, 10^(2)) → radius20
  84. Translate(Rotate[Dilate[pic1, size1], ((2 * pi) * sin(angle1))], Vector[(0, 0), (posX1, posY1)]) → T1
  85. Translate(Rotate[Dilate[pic1, size2], ((2 * pi) * sin(angle2))], Vector[(0, 0), (posX2, posY2)]) → T2
  86. Translate(Rotate[Dilate[pic1, size3], ((2 * pi) * sin(angle3))], Vector[(0, 0), (posX3, posY3)]) → T3
  87. Translate(Rotate[Dilate[pic1, size4], ((2 * pi) * sin(angle4))], Vector[(0, 0), (posX4, posY4)]) → T4
  88. Translate(Rotate[Dilate[pic1, size5], ((2 * pi) * sin(angle5))], Vector[(0, 0), (posX5, posY5)]) → T5
  89. Translate(Rotate[Dilate[pic1, size6], ((2 * pi) * sin(angle6))], Vector[(0, 0), (posX6, posY6)]) → T6
  90. Translate(Rotate[Dilate[pic1, size7], ((2 * pi) * sin(angle7))], Vector[(0, 0), (posX7, posY7)]) → T7
  91. Translate(Rotate[Dilate[pic1, size8], ((2 * pi) * sin(angle8))], Vector[(0, 0), (posX8, posY8)]) → T8
  92. Translate(Rotate[Dilate[pic1, size9], ((2 * pi) * sin(angle9))], Vector[(0, 0), (posX9, posY9)]) → T9
  93. Translate(Rotate[Dilate[pic1, size10], ((2 * pi) * sin(angle10))], Vector[(0, 0), (posX10, posY10)]) → T10
  94. Translate(Rotate[Dilate[pic1, size11], ((2 * pi) * sin(angle11))], Vector[(0, 0), (posX11, posY11)]) → T11
  95. Translate(Rotate[Dilate[pic1, size12], ((2 * pi) * sin(angle12))], Vector[(0, 0), (posX12, posY12)]) → T12
  96. Translate(Rotate[Dilate[pic1, size13], ((2 * pi) * sin(angle13))], Vector[(0, 0), (posX13, posY13)]) → T13
  97. Translate(Rotate[Dilate[pic1, size14], ((2 * pi) * sin(angle14))], Vector[(0, 0), (posX14, posY14)]) → T14
  98. Translate(Rotate[Dilate[pic1, size15], ((2 * pi) * sin(angle15))], Vector[(0, 0), (posX15, posY15)]) → T15
  99. Translate(Rotate[Dilate[pic1, size16], ((2 * pi) * sin(angle16))], Vector[(0, 0), (posX16, posY16)]) → T16
  100. Translate(Rotate[Dilate[pic1, size17], ((2 * pi) * sin(angle17))], Vector[(0, 0), (posX17, posY17)]) → T17
  101. Translate(Rotate[Dilate[pic1, size18], ((2 * pi) * sin(angle18))], Vector[(0, 0), (posX18, posY18)]) → T18
  102. Translate(Rotate[Dilate[pic1, size19], ((2 * pi) * sin(angle19))], Vector[(0, 0), (posX19, posY19)]) → T19
  103. Translate(Rotate[Dilate[pic1, size20], ((2 * pi) * sin(angle20))], Vector[(0, 0), (posX20, posY20)]) → T20

AI 绘图提示语参考

绘制一个冬季场景，包含分形树和雪花图案。使用递归规则生成树枝，树枝的长度和角度通过随机数控制，并随着迭代逐渐缩小。树枝的初始角度和位置也由随机数决定。雪花图案与树枝结合，形成雪花覆盖的冬季森林。整个场景通过参数控制，包括初始角度、长度、宽度和迭代次数，展现分形几何的细节和随机性。
以上提示语为根据原始绘图命令整理而来，仅供参考。