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克莱因瓶非定向曲面演示

克莱因瓶非定向曲面演示
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2022-09-06 11:58
立体几何微积分导数变换参数方程
①本范例演示克莱因瓶这一非定向曲面的三维图形及其特性；②涉及的核心数学概念包括曲面、参数方程、导数及变换；③作图要点提炼自构造摘要，使用了Surface（曲面）、Derivative（导数）、If（条件判断）、Translate（平移）、Mod（模运算）等GeoGebra命令及对象如Ay, Ax, c, b, a, size, H1, G1, F1, Klein1等；④适合高中及以上数学学习者、大学低年级学生及对拓扑学感兴趣的爱好者进行教学演示和概念理解。

GeoGebra 命令及参数

  1. Surface(F1(u, v), G1(u, v), H1(u, v), u, pi, (2 * pi), v, 0, (2 * pi)) → Klein1
  2. Surface(F0(u, v), G0(u, v), H0(u, v), u, 0, pi, v, 0, (2 * pi)) → Klein0
  3. Derivative(F0, x) → dF0x
  4. Derivative(F0, y) → dF0y
  5. Derivative(G0, x) → dG0x
  6. Derivative(G0, y) → dG0y
  7. Derivative(H0, x) → dH0x
  8. Derivative(H0, y) → dH0y
  9. Derivative(F1, x) → dF1x
  10. Derivative(F1, y) → dF1y
  11. Derivative(G1, x) → dG1x
  12. Derivative(G1, y) → dG1y
  13. Derivative(H1, x) → dH1x
  14. Derivative(H1, y) → dH1y
  15. If(Ax < pi, (uN0x(Ax, Ay), uN0y(Ax, Ay), uN0z(Ax, Ay)), If[Ax < (2 * pi), (uN1x(Ax, Ay), uN1y(Ax, Ay), uN1z(Ax, Ay)), If[Ax < (3 * pi), ((-uN0x(Ax, Ay)), (-uN0y(Ax, Ay)), (-uN0z(Ax, Ay))), ((-uN1x(Ax, Ay)), (-uN1y(Ax, Ay)), (-uN1z(Ax, Ay)))]]) → C
  16. If(Ax < pi, Klein0(Ax, Ay), Klein1(Ax, Ay)) → A
  17. Translate(Vector[(0, 0, 0), C], A) → u
  18. Mod(((1 * 4) * pi / n) + Ax, (4 * pi)) → A1
  19. Mod(((2 * 4) * pi / n) + Ax, (4 * pi)) → A2
  20. Mod(((3 * 4) * pi / n) + Ax, (4 * pi)) → A3
  21. Mod(((4 * 4) * pi / n) + Ax, (4 * pi)) → A4
  22. Mod(((5 * 4) * pi / n) + Ax, (4 * pi)) → A5
  23. If(A1 < pi, Klein0(A1, pi / 2), Klein1(A1, pi / 2)) → B1
  24. If(A2 < pi, Klein0(A2, pi / 2), Klein1(A2, pi / 2)) → B2
  25. If(A3 < pi, Klein0(A3, pi / 2), Klein1(A3, pi / 2)) → B3
  26. If(A4 < pi, Klein0(A4, pi / 2), Klein1(A4, pi / 2)) → B4
  27. If(A5 < pi, Klein0(A5, pi / 2), Klein1(A5, pi / 2)) → B5
  28. If(A1 < pi, (uN0x(A1, pi / 2), uN0y(A1, pi / 2), uN0z(A1, pi / 2)), If[A1 < (2 * pi), (uN1x(A1, pi / 2), uN1y(A1, pi / 2), uN1z(A1, pi / 2)), If[A1 < (3 * pi), ((-uN0x(A1, pi / 2)), (-uN0y(A1, pi / 2)), (-uN0z(A1, pi / 2))), ((-uN1x(A1, pi / 2)), (-uN1y(A1, pi / 2)), (-uN1z(A1, pi / 2)))]]) → C1
  29. If(A2 < pi, (uN0x(A2, pi / 2), uN0y(A2, pi / 2), uN0z(A2, pi / 2)), If[A2 < (2 * pi), (uN1x(A2, pi / 2), uN1y(A2, pi / 2), uN1z(A2, pi / 2)), If[A2 < (3 * pi), ((-uN0x(A2, pi / 2)), (-uN0y(A2, pi / 2)), (-uN0z(A2, pi / 2))), ((-uN1x(A2, pi / 2)), (-uN1y(A2, pi / 2)), (-uN1z(A2, pi / 2)))]]) → C2
  30. If(A3 < pi, (uN0x(A3, pi / 2), uN0y(A3, pi / 2), uN0z(A3, pi / 2)), If[A3 < (2 * pi), (uN1x(A3, pi / 2), uN1y(A3, pi / 2), uN1z(A3, pi / 2)), If[A3 < (3 * pi), ((-uN0x(A3, pi / 2)), (-uN0y(A3, pi / 2)), (-uN0z(A3, pi / 2))), ((-uN1x(A3, pi / 2)), (-uN1y(A3, pi / 2)), (-uN1z(A3, pi / 2)))]]) → C3
  31. If(A4 < pi, (uN0x(A4, pi / 2), uN0y(A4, pi / 2), uN0z(A4, pi / 2)), If[A4 < (2 * pi), (uN1x(A4, pi / 2), uN1y(A4, pi / 2), uN1z(A4, pi / 2)), If[A4 < (3 * pi), ((-uN0x(A4, pi / 2)), (-uN0y(A4, pi / 2)), (-uN0z(A4, pi / 2))), ((-uN1x(A4, pi / 2)), (-uN1y(A4, pi / 2)), (-uN1z(A4, pi / 2)))]]) → C4
  32. If(A5 < pi, (uN0x(A5, pi / 2), uN0y(A5, pi / 2), uN0z(A5, pi / 2)), If[A5 < (2 * pi), (uN1x(A5, pi / 2), uN1y(A5, pi / 2), uN1z(A5, pi / 2)), If[A5 < (3 * pi), ((-uN0x(A5, pi / 2)), (-uN0y(A5, pi / 2)), (-uN0z(A5, pi / 2))), ((-uN1x(A5, pi / 2)), (-uN1y(A5, pi / 2)), (-uN1z(A5, pi / 2)))]]) → C5
  33. Translate(Vector[(0, 0, 0), C1], B1) → D1
  34. Translate(Vector[(0, 0, 0), C2], B2) → D2
  35. Translate(Vector[(0, 0, 0), C3], B3) → D3
  36. Translate(Vector[(0, 0, 0), C4], B4) → D4
  37. Translate(Vector[(0, 0, 0), C5], B5) → D5
  38. Mod(((6 * 4) * pi / n) + Ax, (4 * pi)) → A6
  39. Mod(((7 * 4) * pi / n) + Ax, (4 * pi)) → A7
  40. Mod(((8 * 4) * pi / n) + Ax, (4 * pi)) → A8
  41. Mod(((9 * 4) * pi / n) + Ax, (4 * pi)) → A9
  42. Mod(((10 * 4) * pi / n) + Ax, (4 * pi)) → A10
  43. Mod(((11 * 4) * pi / n) + Ax, (4 * pi)) → A11
  44. Mod(((12 * 4) * pi / n) + Ax, (4 * pi)) → A12
  45. Mod(((13 * 4) * pi / n) + Ax, (4 * pi)) → A13
  46. Mod(((14 * 4) * pi / n) + Ax, (4 * pi)) → A14
  47. Mod(((15 * 4) * pi / n) + Ax, (4 * pi)) → A15
  48. If(A6 < pi, Klein0(A6, pi / 2), Klein1(A6, pi / 2)) → B6
  49. If(A7 < pi, Klein0(A7, pi / 2), Klein1(A7, pi / 2)) → B7
  50. If(A8 < pi, Klein0(A8, pi / 2), Klein1(A8, pi / 2)) → B8
  51. If(A9 < pi, Klein0(A9, pi / 2), Klein1(A9, pi / 2)) → B9
  52. If(A10 < pi, Klein0(A10, pi / 2), Klein1(A10, pi / 2)) → B10
  53. If(A11 < pi, Klein0(A11, pi / 2), Klein1(A11, pi / 2)) → B11
  54. If(A12 < pi, Klein0(A12, pi / 2), Klein1(A12, pi / 2)) → B12
  55. If(A13 < pi, Klein0(A13, pi / 2), Klein1(A13, pi / 2)) → B13
  56. If(A14 < pi, Klein0(A14, pi / 2), Klein1(A14, pi / 2)) → B14
  57. If(A15 < pi, Klein0(A15, pi / 2), Klein1(A15, pi / 2)) → B15
  58. If(A6 < pi, (uN0x(A6, pi / 2), uN0y(A6, pi / 2), uN0z(A6, pi / 2)), If[A6 < (2 * pi), (uN1x(A6, pi / 2), uN1y(A6, pi / 2), uN1z(A6, pi / 2)), If[A6 < (3 * pi), ((-uN0x(A6, pi / 2)), (-uN0y(A6, pi / 2)), (-uN0z(A6, pi / 2))), ((-uN1x(A6, pi / 2)), (-uN1y(A6, pi / 2)), (-uN1z(A6, pi / 2)))]]) → C6
  59. If(A7 < pi, (uN0x(A7, pi / 2), uN0y(A7, pi / 2), uN0z(A7, pi / 2)), If[A7 < (2 * pi), (uN1x(A7, pi / 2), uN1y(A7, pi / 2), uN1z(A7, pi / 2)), If[A7 < (3 * pi), ((-uN0x(A7, pi / 2)), (-uN0y(A7, pi / 2)), (-uN0z(A7, pi / 2))), ((-uN1x(A7, pi / 2)), (-uN1y(A7, pi / 2)), (-uN1z(A7, pi / 2)))]]) → C7
  60. If(A8 < pi, (uN0x(A8, pi / 2), uN0y(A8, pi / 2), uN0z(A8, pi / 2)), If[A8 < (2 * pi), (uN1x(A8, pi / 2), uN1y(A8, pi / 2), uN1z(A8, pi / 2)), If[A8 < (3 * pi), ((-uN0x(A8, pi / 2)), (-uN0y(A8, pi / 2)), (-uN0z(A8, pi / 2))), ((-uN1x(A8, pi / 2)), (-uN1y(A8, pi / 2)), (-uN1z(A8, pi / 2)))]]) → C8
  61. If(A9 < pi, (uN0x(A9, pi / 2), uN0y(A9, pi / 2), uN0z(A9, pi / 2)), If[A9 < (2 * pi), (uN1x(A9, pi / 2), uN1y(A9, pi / 2), uN1z(A9, pi / 2)), If[A9 < (3 * pi), ((-uN0x(A9, pi / 2)), (-uN0y(A9, pi / 2)), (-uN0z(A9, pi / 2))), ((-uN1x(A9, pi / 2)), (-uN1y(A9, pi / 2)), (-uN1z(A9, pi / 2)))]]) → C9
  62. If(A10 < pi, (uN0x(A10, pi / 2), uN0y(A10, pi / 2), uN0z(A10, pi / 2)), If[A10 < (2 * pi), (uN1x(A10, pi / 2), uN1y(A10, pi / 2), uN1z(A10, pi / 2)), If[A10 < (3 * pi), ((-uN0x(A10, pi / 2)), (-uN0y(A10, pi / 2)), (-uN0z(A10, pi / 2))), ((-uN1x(A10, pi / 2)), (-uN1y(A10, pi / 2)), (-uN1z(A10, pi / 2)))]]) → C10
  63. If(A11 < pi, (uN0x(A11, pi / 2), uN0y(A11, pi / 2), uN0z(A11, pi / 2)), If[A11 < (2 * pi), (uN1x(A11, pi / 2), uN1y(A11, pi / 2), uN1z(A11, pi / 2)), If[A11 < (3 * pi), ((-uN0x(A11, pi / 2)), (-uN0y(A11, pi / 2)), (-uN0z(A11, pi / 2))), ((-uN1x(A11, pi / 2)), (-uN1y(A11, pi / 2)), (-uN1z(A11, pi / 2)))]]) → C11
  64. If(A12 < pi, (uN0x(A12, pi / 2), uN0y(A12, pi / 2), uN0z(A12, pi / 2)), If[A12 < (2 * pi), (uN1x(A12, pi / 2), uN1y(A12, pi / 2), uN1z(A12, pi / 2)), If[A12 < (3 * pi), ((-uN0x(A12, pi / 2)), (-uN0y(A12, pi / 2)), (-uN0z(A12, pi / 2))), ((-uN1x(A12, pi / 2)), (-uN1y(A12, pi / 2)), (-uN1z(A12, pi / 2)))]]) → C12
  65. If(A13 < pi, (uN0x(A13, pi / 2), uN0y(A13, pi / 2), uN0z(A13, pi / 2)), If[A13 < (2 * pi), (uN1x(A13, pi / 2), uN1y(A13, pi / 2), uN1z(A13, pi / 2)), If[A13 < (3 * pi), ((-uN0x(A13, pi / 2)), (-uN0y(A13, pi / 2)), (-uN0z(A13, pi / 2))), ((-uN1x(A13, pi / 2)), (-uN1y(A13, pi / 2)), (-uN1z(A13, pi / 2)))]]) → C13
  66. If(A14 < pi, (uN0x(A14, pi / 2), uN0y(A14, pi / 2), uN0z(A14, pi / 2)), If[A14 < (2 * pi), (uN1x(A14, pi / 2), uN1y(A14, pi / 2), uN1z(A14, pi / 2)), If[A14 < (3 * pi), ((-uN0x(A14, pi / 2)), (-uN0y(A14, pi / 2)), (-uN0z(A14, pi / 2))), ((-uN1x(A14, pi / 2)), (-uN1y(A14, pi / 2)), (-uN1z(A14, pi / 2)))]]) → C14
  67. If(A15 < pi, (uN0x(A15, pi / 2), uN0y(A15, pi / 2), uN0z(A15, pi / 2)), If[A15 < (2 * pi), (uN1x(A15, pi / 2), uN1y(A15, pi / 2), uN1z(A15, pi / 2)), If[A15 < (3 * pi), ((-uN0x(A15, pi / 2)), (-uN0y(A15, pi / 2)), (-uN0z(A15, pi / 2))), ((-uN1x(A15, pi / 2)), (-uN1y(A15, pi / 2)), (-uN1z(A15, pi / 2)))]]) → C15
  68. Translate(Vector[(0, 0, 0), C6], B6) → D6
  69. Translate(Vector[(0, 0, 0), C7], B7) → D7
  70. Translate(Vector[(0, 0, 0), C8], B8) → D8
  71. Translate(Vector[(0, 0, 0), C9], B9) → D9
  72. Translate(Vector[(0, 0, 0), C10], B10) → D10
  73. Translate(Vector[(0, 0, 0), C11], B11) → D11
  74. Translate(Vector[(0, 0, 0), C12], B12) → D12
  75. Translate(Vector[(0, 0, 0), C13], B13) → D13
  76. Translate(Vector[(0, 0, 0), C14], B14) → D14
  77. Translate(Vector[(0, 0, 0), C15], B15) → D15
  78. Surface((1.01 * F0(u, v)), (1.01 * G0(u, v)), (1.01 * H0(u, v)), u, 0, pi, v, pi / 2 - 0.2, pi / 2 + 0.2) → Klein00
  79. Surface((1.01 * F1(u, v)), (1.01 * G1(u, v)), (1.01 * H1(u, v)), u, pi, (2 * pi), v, pi / 2 - 0.2, pi / 2 + 0.2) → Klein01

AI 绘图提示语参考

请绘制一个三维的Klein瓶，它是一个非定向曲面。图形由参数方程定义的曲面构成，关键对象包括曲面参数方程、导数计算和变换操作。构造顺序为：先定义参数方程，再计算导数，然后通过条件判断和模运算生成不同部分，最后进行平移。整体呈现一个具有拓扑特性的立体图形，适合用于理解曲面与变换的关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。