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同伦变换：罗马曲面与Boy曲面的连续变形

同伦变换：罗马曲面与Boy曲面的连续变形
几何画板及绘图内容加载中，请稍等……
2020-01-07 16:32
数列立体几何变换参数方程滑块动画
① 本范例通过GeoGebra的3D绘图功能，动态演示了从罗马曲面（Roman surface）连续变形到Boy曲面（Boy surface）的同伦过程，直观展示两个拓扑曲面之间的形变关系。② 核心数学概念包括同伦（homotopy）、曲面参数化、拓扑等价以及浸入曲面（immersion）的连续变形，涉及罗马曲面和Boy曲面的经典构造。③ 作图要点：利用GeoGebra的Surface命令创建参数曲面，并通过Sequence命令生成一系列中间曲面帧，配合滑块（slider）控制变形参数；对象列表中包含多个辅助点（A1-A5、B1-B4等）和曲线（l1、l2），用于构建和约束曲面形状，同时使用图像对象（image2、image3、image4）增强可视化效果。④ 适合大学数学专业或高年级本科生学习拓扑学、微分几何时使用，也可作为数学可视化演示的优质案例，帮助理解抽象的同伦概念。

GeoGebra 命令及参数

  1. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, 0, π / 2, v, A1, A2) → B1
  2. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, 0, π / 2, v, A2, A3) → B2
  3. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, 0, π / 2, v, A3, A4) → B3
  4. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, 0, π / 2, v, A4, A5) → B4
  5. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, π / 2, (2 * π / 2), v, A1, A2) → C1
  6. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, π / 2, (2 * π / 2), v, A2, A3) → C2
  7. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, π / 2, (2 * π / 2), v, A3, A4) → C3
  8. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, π / 2, (2 * π / 2), v, A4, A5) → C4
  9. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, π, (3 * π / 2), v, A1, A2) → D1
  10. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, (3 * π / 2), (4 * π / 2), v, A1, A2) → E1
  11. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, (3 * π / 2), (4 * π / 2), v, A2, A3) → E2
  12. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, (3 * π / 2), (4 * π / 2), v, A3, A4) → E3
  13. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, π, (3 * π / 2), v, A2, A3) → D2
  14. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, π, (3 * π / 2), v, A3, A4) → D3
  15. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, π, (3 * π / 2), v, A4, A5) → D4
  16. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, (3 * π / 2), (4 * π / 2), v, A4, A5) → E4
  17. Sequence(CurveCartesian[x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, 0, (2 * π)], v, 0, (2 * π), (2 * π / m)) → l1
  18. Sequence(CurveCartesian[x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), v, 0, (2 * π)], u, 0, (2 * π), (2 * π / m)) → l2