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克莱因瓶与单位法向量及莫比乌斯带

克莱因瓶与单位法向量及莫比乌斯带
几何画板及绘图内容加载中，请稍等……
2019-11-01 12:36
向量立体几何微积分
该范例展示了克莱因瓶及其单位法向量，并附有莫比乌斯带。它涉及解析几何、向量和微积分等核心数学概念。通过Surface、Derivative、Translate等GeoGebra命令，用户可以直观地理解这些几何对象的构造过程。适合高中及以上学生或教师使用，有助于深入理解三维几何和拓扑学的基本概念。

GeoGebra 命令及参数

  1. Surface(F1(u, v), G1(u, v), H1(u, v), u, π, (2 * π), v, 0, (2 * π)) → Klein1
  2. Surface(F0(u, v), G0(u, v), H0(u, v), u, 0, π, v, 0, (2 * π)) → 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 < π, (uN0x(Ax, Ay), uN0y(Ax, Ay), uN0z(Ax, Ay)), If[Ax < (2 * π), (uN1x(Ax, Ay), uN1y(Ax, Ay), uN1z(Ax, Ay)), If[Ax < (3 * π), ((-uN0x(Ax, Ay)), (-uN0y(Ax, Ay)), (-uN0z(Ax, Ay))), ((-uN1x(Ax, Ay)), (-uN1y(Ax, Ay)), (-uN1z(Ax, Ay)))]]) → C
  16. If(Ax < π, Klein0(Ax, Ay), Klein1(Ax, Ay)) → A
  17. Translate(Vector[(0, 0, 0), C], A) → u
  18. Mod((4 * π / n) + Ax, (4 * π)) → A1
  19. Mod((8 * π / n) + Ax, (4 * π)) → A2
  20. Mod((12 * π / n) + Ax, (4 * π)) → A3
  21. Mod((16 * π / n) + Ax, (4 * π)) → A4
  22. Mod((20 * π / n) + Ax, (4 * π)) → A5
  23. If(A1 < π, Klein0(A1, π / 2), Klein1(A1, π / 2)) → B1
  24. If(A2 < π, Klein0(A2, π / 2), Klein1(A2, π / 2)) → B2
  25. If(A3 < π, Klein0(A3, π / 2), Klein1(A3, π / 2)) → B3
  26. If(A4 < π, Klein0(A4, π / 2), Klein1(A4, π / 2)) → B4
  27. If(A5 < π, Klein0(A5, π / 2), Klein1(A5, π / 2)) → B5
  28. If(A1 < π, (uN0x(A1, π / 2), uN0y(A1, π / 2), uN0z(A1, π / 2)), If[A1 < (2 * π), (uN1x(A1, π / 2), uN1y(A1, π / 2), uN1z(A1, π / 2)), If[A1 < (3 * π), ((-uN0x(A1, π / 2)), (-uN0y(A1, π / 2)), (-uN0z(A1, π / 2))), ((-uN1x(A1, π / 2)), (-uN1y(A1, π / 2)), (-uN1z(A1, π / 2)))]]) → C1
  29. If(A2 < π, (uN0x(A2, π / 2), uN0y(A2, π / 2), uN0z(A2, π / 2)), If[A2 < (2 * π), (uN1x(A2, π / 2), uN1y(A2, π / 2), uN1z(A2, π / 2)), If[A2 < (3 * π), ((-uN0x(A2, π / 2)), (-uN0y(A2, π / 2)), (-uN0z(A2, π / 2))), ((-uN1x(A2, π / 2)), (-uN1y(A2, π / 2)), (-uN1z(A2, π / 2)))]]) → C2
  30. If(A3 < π, (uN0x(A3, π / 2), uN0y(A3, π / 2), uN0z(A3, π / 2)), If[A3 < (2 * π), (uN1x(A3, π / 2), uN1y(A3, π / 2), uN1z(A3, π / 2)), If[A3 < (3 * π), ((-uN0x(A3, π / 2)), (-uN0y(A3, π / 2)), (-uN0z(A3, π / 2))), ((-uN1x(A3, π / 2)), (-uN1y(A3, π / 2)), (-uN1z(A3, π / 2)))]]) → C3
  31. If(A4 < π, (uN0x(A4, π / 2), uN0y(A4, π / 2), uN0z(A4, π / 2)), If[A4 < (2 * π), (uN1x(A4, π / 2), uN1y(A4, π / 2), uN1z(A4, π / 2)), If[A4 < (3 * π), ((-uN0x(A4, π / 2)), (-uN0y(A4, π / 2)), (-uN0z(A4, π / 2))), ((-uN1x(A4, π / 2)), (-uN1y(A4, π / 2)), (-uN1z(A4, π / 2)))]]) → C4
  32. If(A5 < π, (uN0x(A5, π / 2), uN0y(A5, π / 2), uN0z(A5, π / 2)), If[A5 < (2 * π), (uN1x(A5, π / 2), uN1y(A5, π / 2), uN1z(A5, π / 2)), If[A5 < (3 * π), ((-uN0x(A5, π / 2)), (-uN0y(A5, π / 2)), (-uN0z(A5, π / 2))), ((-uN1x(A5, π / 2)), (-uN1y(A5, π / 2)), (-uN1z(A5, π / 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((24 * π / n) + Ax, (4 * π)) → A6
  39. Mod((28 * π / n) + Ax, (4 * π)) → A7
  40. Mod((32 * π / n) + Ax, (4 * π)) → A8
  41. Mod((36 * π / n) + Ax, (4 * π)) → A9
  42. Mod((40 * π / n) + Ax, (4 * π)) → A10
  43. Mod((44 * π / n) + Ax, (4 * π)) → A11
  44. Mod((48 * π / n) + Ax, (4 * π)) → A12
  45. Mod((52 * π / n) + Ax, (4 * π)) → A13
  46. Mod((56 * π / n) + Ax, (4 * π)) → A14
  47. Mod((60 * π / n) + Ax, (4 * π)) → A15
  48. Mod((64 * π / n) + Ax, (4 * π)) → A16
  49. Mod((68 * π / n) + Ax, (4 * π)) → A17
  50. Mod((72 * π / n) + Ax, (4 * π)) → A18
  51. Mod((76 * π / n) + Ax, (4 * π)) → A19
  52. Mod((80 * π / n) + Ax, (4 * π)) → A20
  53. Mod((84 * π / n) + Ax, (4 * π)) → A21
  54. Mod((88 * π / n) + Ax, (4 * π)) → A22
  55. Mod((92 * π / n) + Ax, (4 * π)) → A23
  56. Mod((96 * π / n) + Ax, (4 * π)) → A24
  57. Mod((100 * π / n) + Ax, (4 * π)) → A25
  58. If(A6 < π, Klein0(A6, π / 2), Klein1(A6, π / 2)) → B6
  59. If(A7 < π, Klein0(A7, π / 2), Klein1(A7, π / 2)) → B7
  60. If(A8 < π, Klein0(A8, π / 2), Klein1(A8, π / 2)) → B8
  61. If(A9 < π, Klein0(A9, π / 2), Klein1(A9, π / 2)) → B9
  62. If(A10 < π, Klein0(A10, π / 2), Klein1(A10, π / 2)) → B10
  63. If(A11 < π, Klein0(A11, π / 2), Klein1(A11, π / 2)) → B11
  64. If(A12 < π, Klein0(A12, π / 2), Klein1(A12, π / 2)) → B12
  65. If(A13 < π, Klein0(A13, π / 2), Klein1(A13, π / 2)) → B13
  66. If(A14 < π, Klein0(A14, π / 2), Klein1(A14, π / 2)) → B14
  67. If(A15 < π, Klein0(A15, π / 2), Klein1(A15, π / 2)) → B15
  68. If(A16 < π, Klein0(A16, π / 2), Klein1(A16, π / 2)) → B16
  69. If(A17 < π, Klein0(A17, π / 2), Klein1(A17, π / 2)) → B17
  70. If(A18 < π, Klein0(A18, π / 2), Klein1(A18, π / 2)) → B18
  71. If(A19 < π, Klein0(A19, π / 2), Klein1(A19, π / 2)) → B19
  72. If(A20 < π, Klein0(A20, π / 2), Klein1(A20, π / 2)) → B20
  73. If(A21 < π, Klein0(A21, π / 2), Klein1(A21, π / 2)) → B21
  74. If(A22 < π, Klein0(A22, π / 2), Klein1(A22, π / 2)) → B22
  75. If(A23 < π, Klein0(A23, π / 2), Klein1(A23, π / 2)) → B23
  76. If(A24 < π, Klein0(A24, π / 2), Klein1(A24, π / 2)) → B24
  77. If(A25 < π, Klein0(A25, π / 2), Klein1(A25, π / 2)) → B25
  78. If(A6 < π, (uN0x(A6, π / 2), uN0y(A6, π / 2), uN0z(A6, π / 2)), If[A6 < (2 * π), (uN1x(A6, π / 2), uN1y(A6, π / 2), uN1z(A6, π / 2)), If[A6 < (3 * π), ((-uN0x(A6, π / 2)), (-uN0y(A6, π / 2)), (-uN0z(A6, π / 2))), ((-uN1x(A6, π / 2)), (-uN1y(A6, π / 2)), (-uN1z(A6, π / 2)))]]) → C6
  79. If(A7 < π, (uN0x(A7, π / 2), uN0y(A7, π / 2), uN0z(A7, π / 2)), If[A7 < (2 * π), (uN1x(A7, π / 2), uN1y(A7, π / 2), uN1z(A7, π / 2)), If[A7 < (3 * π), ((-uN0x(A7, π / 2)), (-uN0y(A7, π / 2)), (-uN0z(A7, π / 2))), ((-uN1x(A7, π / 2)), (-uN1y(A7, π / 2)), (-uN1z(A7, π / 2)))]]) → C7
  80. If(A8 < π, (uN0x(A8, π / 2), uN0y(A8, π / 2), uN0z(A8, π / 2)), If[A8 < (2 * π), (uN1x(A8, π / 2), uN1y(A8, π / 2), uN1z(A8, π / 2)), If[A8 < (3 * π), ((-uN0x(A8, π / 2)), (-uN0y(A8, π / 2)), (-uN0z(A8, π / 2))), ((-uN1x(A8, π / 2)), (-uN1y(A8, π / 2)), (-uN1z(A8, π / 2)))]]) → C8
  81. If(A9 < π, (uN0x(A9, π / 2), uN0y(A9, π / 2), uN0z(A9, π / 2)), If[A9 < (2 * π), (uN1x(A9, π / 2), uN1y(A9, π / 2), uN1z(A9, π / 2)), If[A9 < (3 * π), ((-uN0x(A9, π / 2)), (-uN0y(A9, π / 2)), (-uN0z(A9, π / 2))), ((-uN1x(A9, π / 2)), (-uN1y(A9, π / 2)), (-uN1z(A9, π / 2)))]]) → C9
  82. If(A10 < π, (uN0x(A10, π / 2), uN0y(A10, π / 2), uN0z(A10, π / 2)), If[A10 < (2 * π), (uN1x(A10, π / 2), uN1y(A10, π / 2), uN1z(A10, π / 2)), If[A10 < (3 * π), ((-uN0x(A10, π / 2)), (-uN0y(A10, π / 2)), (-uN0z(A10, π / 2))), ((-uN1x(A10, π / 2)), (-uN1y(A10, π / 2)), (-uN1z(A10, π / 2)))]]) → C10
  83. If(A11 < π, (uN0x(A11, π / 2), uN0y(A11, π / 2), uN0z(A11, π / 2)), If[A11 < (2 * π), (uN1x(A11, π / 2), uN1y(A11, π / 2), uN1z(A11, π / 2)), If[A11 < (3 * π), ((-uN0x(A11, π / 2)), (-uN0y(A11, π / 2)), (-uN0z(A11, π / 2))), ((-uN1x(A11, π / 2)), (-uN1y(A11, π / 2)), (-uN1z(A11, π / 2)))]]) → C11
  84. If(A12 < π, (uN0x(A12, π / 2), uN0y(A12, π / 2), uN0z(A12, π / 2)), If[A12 < (2 * π), (uN1x(A12, π / 2), uN1y(A12, π / 2), uN1z(A12, π / 2)), If[A12 < (3 * π), ((-uN0x(A12, π / 2)), (-uN0y(A12, π / 2)), (-uN0z(A12, π / 2))), ((-uN1x(A12, π / 2)), (-uN1y(A12, π / 2)), (-uN1z(A12, π / 2)))]]) → C12
  85. If(A13 < π, (uN0x(A13, π / 2), uN0y(A13, π / 2), uN0z(A13, π / 2)), If[A13 < (2 * π), (uN1x(A13, π / 2), uN1y(A13, π / 2), uN1z(A13, π / 2)), If[A13 < (3 * π), ((-uN0x(A13, π / 2)), (-uN0y(A13, π / 2)), (-uN0z(A13, π / 2))), ((-uN1x(A13, π / 2)), (-uN1y(A13, π / 2)), (-uN1z(A13, π / 2)))]]) → C13
  86. If(A14 < π, (uN0x(A14, π / 2), uN0y(A14, π / 2), uN0z(A14, π / 2)), If[A14 < (2 * π), (uN1x(A14, π / 2), uN1y(A14, π / 2), uN1z(A14, π / 2)), If[A14 < (3 * π), ((-uN0x(A14, π / 2)), (-uN0y(A14, π / 2)), (-uN0z(A14, π / 2))), ((-uN1x(A14, π / 2)), (-uN1y(A14, π / 2)), (-uN1z(A14, π / 2)))]]) → C14
  87. If(A15 < π, (uN0x(A15, π / 2), uN0y(A15, π / 2), uN0z(A15, π / 2)), If[A15 < (2 * π), (uN1x(A15, π / 2), uN1y(A15, π / 2), uN1z(A15, π / 2)), If[A15 < (3 * π), ((-uN0x(A15, π / 2)), (-uN0y(A15, π / 2)), (-uN0z(A15, π / 2))), ((-uN1x(A15, π / 2)), (-uN1y(A15, π / 2)), (-uN1z(A15, π / 2)))]]) → C15
  88. If(A16 < π, (uN0x(A16, π / 2), uN0y(A16, π / 2), uN0z(A16, π / 2)), If[A16 < (2 * π), (uN1x(A16, π / 2), uN1y(A16, π / 2), uN1z(A16, π / 2)), If[A16 < (3 * π), ((-uN0x(A16, π / 2)), (-uN0y(A16, π / 2)), (-uN0z(A16, π / 2))), ((-uN1x(A16, π / 2)), (-uN1y(A16, π / 2)), (-uN1z(A16, π / 2)))]]) → C16
  89. If(A17 < π, (uN0x(A17, π / 2), uN0y(A17, π / 2), uN0z(A17, π / 2)), If[A17 < (2 * π), (uN1x(A17, π / 2), uN1y(A17, π / 2), uN1z(A17, π / 2)), If[A17 < (3 * π), ((-uN0x(A17, π / 2)), (-uN0y(A17, π / 2)), (-uN0z(A17, π / 2))), ((-uN1x(A17, π / 2)), (-uN1y(A17, π / 2)), (-uN1z(A17, π / 2)))]]) → C17
  90. If(A18 < π, (uN0x(A18, π / 2), uN0y(A18, π / 2), uN0z(A18, π / 2)), If[A18 < (2 * π), (uN1x(A18, π / 2), uN1y(A18, π / 2), uN1z(A18, π / 2)), If[A18 < (3 * π), ((-uN0x(A18, π / 2)), (-uN0y(A18, π / 2)), (-uN0z(A18, π / 2))), ((-uN1x(A18, π / 2)), (-uN1y(A18, π / 2)), (-uN1z(A18, π / 2)))]]) → C18
  91. If(A19 < π, (uN0x(A19, π / 2), uN0y(A19, π / 2), uN0z(A19, π / 2)), If[A19 < (2 * π), (uN1x(A19, π / 2), uN1y(A19, π / 2), uN1z(A19, π / 2)), If[A19 < (3 * π), ((-uN0x(A19, π / 2)), (-uN0y(A19, π / 2)), (-uN0z(A19, π / 2))), ((-uN1x(A19, π / 2)), (-uN1y(A19, π / 2)), (-uN1z(A19, π / 2)))]]) → C19
  92. If(A20 < π, (uN0x(A20, π / 2), uN0y(A20, π / 2), uN0z(A20, π / 2)), If[A20 < (2 * π), (uN1x(A20, π / 2), uN1y(A20, π / 2), uN1z(A20, π / 2)), If[A20 < (3 * π), ((-uN0x(A20, π / 2)), (-uN0y(A20, π / 2)), (-uN0z(A20, π / 2))), ((-uN1x(A20, π / 2)), (-uN1y(A20, π / 2)), (-uN1z(A20, π / 2)))]]) → C20
  93. If(A21 < π, (uN0x(A21, π / 2), uN0y(A21, π / 2), uN0z(A21, π / 2)), If[A21 < (2 * π), (uN1x(A21, π / 2), uN1y(A21, π / 2), uN1z(A21, π / 2)), If[A21 < (3 * π), ((-uN0x(A21, π / 2)), (-uN0y(A21, π / 2)), (-uN0z(A21, π / 2))), ((-uN1x(A21, π / 2)), (-uN1y(A21, π / 2)), (-uN1z(A21, π / 2)))]]) → C21
  94. If(A22 < π, (uN0x(A22, π / 2), uN0y(A22, π / 2), uN0z(A22, π / 2)), If[A22 < (2 * π), (uN1x(A22, π / 2), uN1y(A22, π / 2), uN1z(A22, π / 2)), If[A22 < (3 * π), ((-uN0x(A22, π / 2)), (-uN0y(A22, π / 2)), (-uN0z(A22, π / 2))), ((-uN1x(A22, π / 2)), (-uN1y(A22, π / 2)), (-uN1z(A22, π / 2)))]]) → C22
  95. If(A23 < π, (uN0x(A23, π / 2), uN0y(A23, π / 2), uN0z(A23, π / 2)), If[A23 < (2 * π), (uN1x(A23, π / 2), uN1y(A23, π / 2), uN1z(A23, π / 2)), If[A23 < (3 * π), ((-uN0x(A23, π / 2)), (-uN0y(A23, π / 2)), (-uN0z(A23, π / 2))), ((-uN1x(A23, π / 2)), (-uN1y(A23, π / 2)), (-uN1z(A23, π / 2)))]]) → C23
  96. If(A24 < π, (uN0x(A24, π / 2), uN0y(A24, π / 2), uN0z(A24, π / 2)), If[A24 < (2 * π), (uN1x(A24, π / 2), uN1y(A24, π / 2), uN1z(A24, π / 2)), If[A24 < (3 * π), ((-uN0x(A24, π / 2)), (-uN0y(A24, π / 2)), (-uN0z(A24, π / 2))), ((-uN1x(A24, π / 2)), (-uN1y(A24, π / 2)), (-uN1z(A24, π / 2)))]]) → C24
  97. If(A25 < π, (uN0x(A25, π / 2), uN0y(A25, π / 2), uN0z(A25, π / 2)), If[A25 < (2 * π), (uN1x(A25, π / 2), uN1y(A25, π / 2), uN1z(A25, π / 2)), If[A25 < (3 * π), ((-uN0x(A25, π / 2)), (-uN0y(A25, π / 2)), (-uN0z(A25, π / 2))), ((-uN1x(A25, π / 2)), (-uN1y(A25, π / 2)), (-uN1z(A25, π / 2)))]]) → C25
  98. Translate(Vector[(0, 0, 0), C6], B6) → D6
  99. Translate(Vector[(0, 0, 0), C7], B7) → D7
  100. Translate(Vector[(0, 0, 0), C8], B8) → D8
  101. Translate(Vector[(0, 0, 0), C9], B9) → D9
  102. Translate(Vector[(0, 0, 0), C10], B10) → D10
  103. Translate(Vector[(0, 0, 0), C11], B11) → D11
  104. Translate(Vector[(0, 0, 0), C12], B12) → D12
  105. Translate(Vector[(0, 0, 0), C13], B13) → D13
  106. Translate(Vector[(0, 0, 0), C14], B14) → D14
  107. Translate(Vector[(0, 0, 0), C15], B15) → D15
  108. Translate(Vector[(0, 0, 0), C16], B16) → D16
  109. Translate(Vector[(0, 0, 0), C17], B17) → D17
  110. Translate(Vector[(0, 0, 0), C18], B18) → D18
  111. Translate(Vector[(0, 0, 0), C19], B19) → D19
  112. Translate(Vector[(0, 0, 0), C20], B20) → D20
  113. Translate(Vector[(0, 0, 0), C21], B21) → D21
  114. Translate(Vector[(0, 0, 0), C22], B22) → D22
  115. Translate(Vector[(0, 0, 0), C23], B23) → D23
  116. Translate(Vector[(0, 0, 0), C24], B24) → D24
  117. Translate(Vector[(0, 0, 0), C25], B25) → D25
  118. Surface((1.01 * F0(u, v)), (1.01 * G0(u, v)), (1.01 * H0(u, v)), u, 0, π, v, π / 2 - 0.2, π / 2 + 0.2) → Klein00
  119. Surface((1.01 * F1(u, v)), (1.01 * G1(u, v)), (1.01 * H1(u, v)), u, π, (2 * π), v, π / 2 - 0.2, π / 2 + 0.2) → Klein01

AI 绘图提示语参考

绘制一个克莱因瓶的三维曲面，该曲面由两段参数化表面拼接而成，并沿其表面分布单位法向量箭头。同时，在克莱因瓶的特定截线上，等距放置多个单位法向量箭头，形成类似莫比乌斯带的环绕效果。整体图形展示克莱因瓶的拓扑结构，法向量方向随曲面位置变化，体现不可定向性。
以上提示语为根据原始绘图命令整理而来，仅供参考。