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立方体错觉

立方体错觉
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2018-11-19 12:48
多边形向量立体几何变换
①本范例演示了一个立方体在二维平面上的错觉效果；②涉及的核心数学概念包括向量、多边形构造、二维变换和序列生成；③作图要点使用了Vector、Sequence、Polygon、Rotate等GeoGebra命令和对象；④适合高中阶段学习立体几何和二维向量变换的学生使用。

GeoGebra 命令及参数

  1. Vector((0, 0), A) → u
  2. Vector((0, 0), B) → v
  3. Sequence(Rotate[(cos(((i / n * 2) * π)), sin(((i / n * 2) * π))), π / 6, (0, 0)], i, 1, n) → Pts
  4. Sequence(Rotate[(((t - 1) * cos(((i / n * 2) * π))), ((t - 1) * sin(((i / n * 2) * π)))), π / 6, (0, 0)], i, 1, n) → PtsC
  5. Polygon(Element[Pts, 2], Element[Pts, 3], Element[Pts, 4], (0, 0), Element[Pts, 2]) → C1, f, g, h, i, j
  6. Rotate(C1, (2 * π / 3), (0, 0)) → L1
  7. Rotate(C1, ((2 * 2) * π / 3), (0, 0)) → L2
  8. Rotate(C1, ((3 * 2) * π / 3), (0, 0)) → L3
  9. Polygon(Element[PtsC, 3], Element[PtsC, 4], Element[PtsC, 5], (0, 0), Element[PtsC, 3]) → C2, k, l, m, p, q
  10. Rotate(C2, (2 * π / 3), (0, 0)) → M1
  11. Rotate(C2, ((2 * 2) * π / 3), (0, 0)) → M2
  12. Rotate(C2, ((3 * 2) * π / 3), (0, 0)) → M3
  13. Polygon(Element[PtsC, 2], Element[PtsC, 3], Element[PtsC, 4], (0, 0), Element[PtsC, 2]) → C3, r, s, a, b, c
  14. Rotate(C3, (2 * π / 3), (0, 0)) → N1
  15. Rotate(C3, ((2 * 2) * π / 3), (0, 0)) → N2
  16. Rotate(C3, ((3 * 2) * π / 3), (0, 0)) → N3
  17. Sequence(Sequence[Translate[M1, Vector[(i * u) + (j * v)]], i, 1, size], j, 1, size) → m_1
  18. Sequence(Sequence[Translate[M2, Vector[(i * u) + (j * v)]], i, 1, size], j, 1, size) → m_2
  19. Sequence(Sequence[Translate[M3, Vector[(i * u) + (j * v)]], i, 1, size], j, 1, size) → m_3
  20. Sequence(Sequence[Translate[L1, Vector[(i * u) + (j * v)]], i, 1, size], j, 1, size) → l_1
  21. Sequence(Sequence[Translate[L2, Vector[(i * u) + (j * v)]], i, 1, size], j, 1, size) → l_2
  22. Sequence(Sequence[Translate[L3, Vector[(i * u) + (j * v)]], i, 1, size], j, 1, size) → l_3
  23. Sequence(Sequence[Translate[N1, Vector[(i * u) + (j * v)]], i, 1, size], j, 1, size) → n_1
  24. Sequence(Sequence[Translate[N2, Vector[(i * u) + (j * v)]], i, 1, size], j, 1, size) → n_2
  25. Sequence(Sequence[Translate[N3, Vector[(i * u) + (j * v)]], i, 1, size], j, 1, size) → n_3