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分形曲线4

分形曲线4
几何画板及绘图内容加载中，请稍等……
2019-10-29 07:25
函数三角函数变换几何作图
①本范例演示了一个复杂的分形曲线图形，通过迭代变换和递归构造生成；②涉及的核心数学概念包括分形几何、几何变换、对称性、函数以及标准差；③作图要点——从下方『构造摘要』提炼用到了PolyLine（多边形）、Rotate（旋转）、Translate（平移）等GeoGebra命令和对象；④适合高中及以上数学学习者，用于理解分形图案的生成原理和几何变换的应用。

GeoGebra 命令及参数

  1. PolyLine(A, G, H, F) → f
  2. Rotate(PolyLine[Dilate[A, 1 / sqrt(2), A], Dilate[G, 1 / sqrt(2), A], Dilate[H, 1 / sqrt(2), A], Dilate[F, 1 / sqrt(2), A]], π / 4, A) → A1
  3. Translate(Rotate[PolyLine[Dilate[A, 0.5 / sqrt(2), A], Dilate[G, 0.5 / sqrt(2), A], Dilate[H, 0.5 / sqrt(2), A], Dilate[F, 0.5 / sqrt(2), A]], ((-π)) / 4, A], B - A) → B1
  4. Rotate(B1, π / 2, C) → C1
  5. Rotate(C1, ((-π)) / 2, D) → D1
  6. Rotate(Translate[B1, F - B], ((-π)) / 2, F) → E1
  7. Rotate(Dilate[{A1, B1, C1, D1, E1}, 1 / sqrt(2), A], π / 4, A) → A2
  8. Translate(Rotate[Dilate[{A1, B1, C1, D1, E1}, 0.5 / sqrt(2), A], ((-π)) / 4, A], B - A) → B2
  9. Rotate(B2, π / 2, C) → C2
  10. Rotate(C2, ((-π)) / 2, D) → D2
  11. Rotate(Translate[B2, F - B], ((-π)) / 2, F) → E2
  12. Rotate(Dilate[{A2, B2, C2, D2, E2}, 1 / sqrt(2), A], π / 4, A) → A3
  13. Translate(Rotate[Dilate[{A2, B2, C2, D2, E2}, 0.500000000000002 / sqrt(2), A], ((-π)) / 4, A], B - A) → B3
  14. Rotate(B3, π / 2, C) → C3
  15. Rotate(C3, ((-π)) / 2, D) → D3
  16. Rotate(Translate[B3, F - B], ((-π)) / 2, F) → E3
  17. Rotate(Dilate[{A3, B3, C3, D3, E3}, 1 / sqrt(2), A], π / 4, A) → A4
  18. Translate(Rotate[Dilate[{A3, B3, C3, D3, E3}, 0.500000000000004 / sqrt(2), A], ((-π)) / 4, A], B - A) → B4
  19. Rotate(B4, π / 2, C) → C4
  20. Rotate(C4, ((-π)) / 2, D) → D4
  21. Rotate(Translate[B4, F - B], ((-π)) / 2, F) → E4
  22. Rotate(Dilate[{A4, B4, C4, D4, E4}, 1 / sqrt(2), A], π / 4, A) → A5
  23. Translate(Rotate[Dilate[{A4, B4, C4, D4, E4}, 0.500000000000006 / sqrt(2), A], ((-π)) / 4, A], B - A) → B5
  24. Rotate(B5, π / 2, C) → C5
  25. Rotate(C5, ((-π)) / 2, D) → D5
  26. Rotate(Translate[B5, F - B], ((-π)) / 2, F) → E5