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Jeener's Klein Surface

Jeener's Klein Surface
几何画板及绘图内容加载中，请稍等……
2020-02-02 19:12
函数立体几何微积分参数方程几何作图
所属专题 Color 3D surfaces
①本范例演示了Jeener's Klein Surface的图形及其连续性；②涉及的核心数学概念包括曲面、连续性、正态分布、构造、同余、分段函数；③作图要点使用了Surface、n、a、b、c、X、Y、Z、O、A1、m、g、A2、A3、A4、A5、A6、A7、A8、A9、A10、A11、A12、A13、A14、A15、A16、A17、A18、A19、A20、A21、A22、A23、A24、A25、A26、B1、B2、B3、B4等GeoGebra命令/对象；④适合高中及以上学段学习曲面构造与高等数学概念。

GeoGebra 命令及参数

  1. Surface(X(u, v), Y(u, v), Z(u, v), u, 0, (2 * π), v, 0, (2 * π)) → O
  2. Surface(X(u, v), Y(u, v), Z(u, v), u, A1, A2, v, 0, (2 * π)) → B1
  3. Surface(X(u, v), Y(u, v), Z(u, v), u, A2, A3, v, 0, (2 * π)) → B2
  4. Surface(X(u, v), Y(u, v), Z(u, v), u, A3, A4, v, 0, (2 * π)) → B3
  5. Surface(X(u, v), Y(u, v), Z(u, v), u, A4, A5, v, 0, (2 * π)) → B4
  6. Surface(X(u, v), Y(u, v), Z(u, v), u, A5, A6, v, 0, (2 * π)) → B5
  7. Surface(X(u, v), Y(u, v), Z(u, v), u, A6, A7, v, 0, (2 * π)) → B6
  8. Surface(X(u, v), Y(u, v), Z(u, v), u, A7, A8, v, 0, (2 * π)) → B7
  9. Surface(X(u, v), Y(u, v), Z(u, v), u, A8, A9, v, 0, (2 * π)) → B8
  10. Surface(X(u, v), Y(u, v), Z(u, v), u, A9, A10, v, 0, (2 * π)) → B9
  11. Surface(X(u, v), Y(u, v), Z(u, v), u, A10, A11, v, 0, (2 * π)) → B10
  12. Surface(X(u, v), Y(u, v), Z(u, v), u, A11, A12, v, 0, (2 * π)) → B11
  13. Surface(X(u, v), Y(u, v), Z(u, v), u, A12, A13, v, 0, (2 * π)) → B12
  14. Surface(X(u, v), Y(u, v), Z(u, v), u, A13, A14, v, 0, (2 * π)) → B13
  15. Surface(X(u, v), Y(u, v), Z(u, v), u, A14, A15, v, 0, (2 * π)) → B14
  16. Surface(X(u, v), Y(u, v), Z(u, v), u, A15, A16, v, 0, (2 * π)) → B15
  17. Surface(X(u, v), Y(u, v), Z(u, v), u, A16, A17, v, 0, (2 * π)) → B16
  18. Surface(X(u, v), Y(u, v), Z(u, v), u, A17, A18, v, 0, (2 * π)) → B17
  19. Surface(X(u, v), Y(u, v), Z(u, v), u, A18, A19, v, 0, (2 * π)) → B18
  20. Surface(X(u, v), Y(u, v), Z(u, v), u, A19, A20, v, 0, (2 * π)) → B19
  21. Surface(X(u, v), Y(u, v), Z(u, v), u, A20, A21, v, 0, (2 * π)) → B20
  22. Surface(X(u, v), Y(u, v), Z(u, v), u, A21, A22, v, 0, (2 * π)) → B21
  23. Surface(X(u, v), Y(u, v), Z(u, v), u, A22, A23, v, 0, (2 * π)) → B22
  24. Surface(X(u, v), Y(u, v), Z(u, v), u, A23, A24, v, 0, (2 * π)) → B23
  25. Surface(X(u, v), Y(u, v), Z(u, v), u, A24, A25, v, 0, (2 * π)) → B24
  26. Surface(X(u, v), Y(u, v), Z(u, v), u, A25, A26, v, 0, (2 * π)) → B25
  27. Surface(X(u, v), Y(u, v), Z(u, v), u, A26, A27, v, 0, (2 * π)) → B26
  28. Surface(X(u, v), Y(u, v), Z(u, v), u, A27, A28, v, 0, (2 * π)) → B27
  29. Surface(X(u, v), Y(u, v), Z(u, v), u, A28, A29, v, 0, (2 * π)) → B28
  30. Surface(X(u, v), Y(u, v), Z(u, v), u, A29, A30, v, 0, (2 * π)) → B29
  31. Surface(X(u, v), Y(u, v), Z(u, v), u, A30, A31, v, 0, (2 * π)) → B30
  32. Surface(X(u, v), Y(u, v), Z(u, v), u, A31, A32, v, 0, (2 * π)) → B31
  33. Surface(X(u, v), Y(u, v), Z(u, v), u, A32, A33, v, 0, (2 * π)) → B32
  34. Surface(X(u, v), Y(u, v), Z(u, v), u, A33, A34, v, 0, (2 * π)) → B33
  35. Surface(X(u, v), Y(u, v), Z(u, v), u, A34, A35, v, 0, (2 * π)) → B34
  36. Surface(X(u, v), Y(u, v), Z(u, v), u, A35, A36, v, 0, (2 * π)) → B35
  37. Surface(X(u, v), Y(u, v), Z(u, v), u, A36, A37, v, 0, (2 * π)) → B36
  38. Surface(X(u, v), Y(u, v), Z(u, v), u, A37, A38, v, 0, (2 * π)) → B37
  39. Surface(X(u, v), Y(u, v), Z(u, v), u, A38, A39, v, 0, (2 * π)) → B38
  40. Surface(X(u, v), Y(u, v), Z(u, v), u, A39, A40, v, 0, (2 * π)) → B39
  41. Surface(X(u, v), Y(u, v), Z(u, v), u, A40, A41, v, 0, (2 * π)) → B40
  42. Surface(X(u, v), Y(u, v), Z(u, v), u, A41, A42, v, 0, (2 * π)) → B41
  43. Surface(X(u, v), Y(u, v), Z(u, v), u, A42, A43, v, 0, (2 * π)) → B42
  44. Surface(X(u, v), Y(u, v), Z(u, v), u, A43, A44, v, 0, (2 * π)) → B43
  45. Surface(X(u, v), Y(u, v), Z(u, v), u, A44, A45, v, 0, (2 * π)) → B44
  46. Surface(X(u, v), Y(u, v), Z(u, v), u, A45, A46, v, 0, (2 * π)) → B45
  47. Surface(X(u, v), Y(u, v), Z(u, v), u, A46, A47, v, 0, (2 * π)) → B46
  48. Surface(X(u, v), Y(u, v), Z(u, v), u, A47, A48, v, 0, (2 * π)) → B47
  49. Surface(X(u, v), Y(u, v), Z(u, v), u, A48, A49, v, 0, (2 * π)) → B48
  50. Surface(X(u, v), Y(u, v), Z(u, v), u, A49, A50, v, 0, (2 * π)) → B49
  51. Surface(X(u, v), Y(u, v), Z(u, v), u, A50, A51, v, 0, (2 * π)) → B50
  52. Surface(X(u, v), Y(u, v), Z(u, v), u, A51, A52, v, 0, (2 * π)) → B51
  53. Surface(X(u, v), Y(u, v), Z(u, v), u, A52, A53, v, 0, (2 * π)) → B52

AI 绘图提示语参考

绘制一个由参数方程定义的闭合曲面，该曲面在参数域上连续变化，并利用正态分布与分段函数构造其形状。整个曲面由多个沿参数方向分割的曲面片拼接而成，每个曲面片覆盖参数区间的一个子段，且所有曲面片在边界处保持连续。曲面整体呈现为一种具有复杂褶皱和对称性的三维形态，类似于克莱因曲面的变体。构造时先定义完整的曲面，再将其按参数区间细分为若干部分，以展示其局部结构与连续性。
以上提示语为根据原始绘图命令整理而来，仅供参考。