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莫比乌斯带3D视图

莫比乌斯带3D视图
几何画板及绘图内容加载中，请稍等……
2015-06-28 10:05
函数解析几何立体几何参数方程
①这个范例演示了莫比乌斯带的3D视图及其特性；②涉及的核心数学概念包括曲面、参数方程、解析几何；③作图要点使用了Surface（曲面）和CurveCartesian（笛卡尔曲线）命令，以及多个点对象（a, A1-A17, B1-B11, C1-C17）；④适合高中及以上学生学习和理解空间几何与参数方程的应用。

GeoGebra 命令及参数

  1. Surface(((3 + (v * cos(u / 2))) * cos(u)), ((3 + (v * cos(u / 2))) * sin(u)), (v * sin(u / 2)), u, 0, 6.283185307179586, v, -1, 1) → a
  2. CurveCartesian(((3 + (A1 * cos(t / 2))) * cos(t)), ((3 + (A1 * cos(t / 2))) * sin(t)), (A1 * sin(t / 2)), t, 0, 6.283185307179586) → B1
  3. CurveCartesian(((3 + (A2 * cos(t / 2))) * cos(t)), ((3 + (A2 * cos(t / 2))) * sin(t)), (A2 * sin(t / 2)), t, 0, 6.2831853071796) → B2
  4. CurveCartesian(((3 + (A3 * cos(t / 2))) * cos(t)), ((3 + (A3 * cos(t / 2))) * sin(t)), (A3 * sin(t / 2)), t, 0, 6.2831853071796) → B3
  5. CurveCartesian(((3 + (A4 * cos(t / 2))) * cos(t)), ((3 + (A4 * cos(t / 2))) * sin(t)), (A4 * sin(t / 2)), t, 0, 6.2831853071796) → B4
  6. CurveCartesian(((3 + (A5 * cos(t / 2))) * cos(t)), ((3 + (A5 * cos(t / 2))) * sin(t)), (A5 * sin(t / 2)), t, 0, 6.2831853071796) → B5
  7. CurveCartesian(((3 + (A6 * cos(t / 2))) * cos(t)), ((3 + (A6 * cos(t / 2))) * sin(t)), (A6 * sin(t / 2)), t, 0, 6.2831853071796) → B6
  8. CurveCartesian(((3 + (A7 * cos(t / 2))) * cos(t)), ((3 + (A7 * cos(t / 2))) * sin(t)), (A7 * sin(t / 2)), t, 0, 6.2831853071796) → B7
  9. CurveCartesian(((3 + (A8 * cos(t / 2))) * cos(t)), ((3 + (A8 * cos(t / 2))) * sin(t)), (A8 * sin(t / 2)), t, 0, 6.2831853071796) → B8
  10. CurveCartesian(((3 + (A9 * cos(t / 2))) * cos(t)), ((3 + (A9 * cos(t / 2))) * sin(t)), (A9 * sin(t / 2)), t, 0, 6.2831853071796) → B9
  11. CurveCartesian(((3 + (A10 * cos(t / 2))) * cos(t)), ((3 + (A10 * cos(t / 2))) * sin(t)), (A10 * sin(t / 2)), t, 0, 6.2831853071796) → B10
  12. CurveCartesian(((3 + (A11 * cos(t / 2))) * cos(t)), ((3 + (A11 * cos(t / 2))) * sin(t)), (A11 * sin(t / 2)), t, 0, 6.2831853071796) → B11
  13. CurveCartesian(((3 + (t * cos(C1 / 2))) * cos(C1)), ((3 + (t * cos(C1 / 2))) * sin(C1)), (t * sin(C1 / 2)), t, -1, 1) → D1
  14. CurveCartesian(((3 + (t * cos(C2 / 2))) * cos(C2)), ((3 + (t * cos(C2 / 2))) * sin(C2)), (t * sin(C2 / 2)), t, -1, 1) → D2
  15. CurveCartesian(((3 + (t * cos(C3 / 2))) * cos(C3)), ((3 + (t * cos(C3 / 2))) * sin(C3)), (t * sin(C3 / 2)), t, -1, 1) → D3
  16. CurveCartesian(((3 + (t * cos(C4 / 2))) * cos(C4)), ((3 + (t * cos(C4 / 2))) * sin(C4)), (t * sin(C4 / 2)), t, -1, 1) → D4
  17. CurveCartesian(((3 + (t * cos(C5 / 2))) * cos(C5)), ((3 + (t * cos(C5 / 2))) * sin(C5)), (t * sin(C5 / 2)), t, -1, 1) → D5
  18. CurveCartesian(((3 + (t * cos(C6 / 2))) * cos(C6)), ((3 + (t * cos(C6 / 2))) * sin(C6)), (t * sin(C6 / 2)), t, -1, 1) → D6
  19. CurveCartesian(((3 + (t * cos(C7 / 2))) * cos(C7)), ((3 + (t * cos(C7 / 2))) * sin(C7)), (t * sin(C7 / 2)), t, -1, 1) → D7
  20. CurveCartesian(((3 + (t * cos(C8 / 2))) * cos(C8)), ((3 + (t * cos(C8 / 2))) * sin(C8)), (t * sin(C8 / 2)), t, -1, 1) → D8
  21. CurveCartesian(((3 + (t * cos(C9 / 2))) * cos(C9)), ((3 + (t * cos(C9 / 2))) * sin(C9)), (t * sin(C9 / 2)), t, -1, 1) → D9
  22. CurveCartesian(((3 + (t * cos(C10 / 2))) * cos(C10)), ((3 + (t * cos(C10 / 2))) * sin(C10)), (t * sin(C10 / 2)), t, -1, 1) → D10
  23. CurveCartesian(((3 + (t * cos(C11 / 2))) * cos(C11)), ((3 + (t * cos(C11 / 2))) * sin(C11)), (t * sin(C11 / 2)), t, -1, 1) → D11
  24. CurveCartesian(((3 + (t * cos(C12 / 2))) * cos(C12)), ((3 + (t * cos(C12 / 2))) * sin(C12)), (t * sin(C12 / 2)), t, -1, 1) → D12
  25. CurveCartesian(((3 + (t * cos(C13 / 2))) * cos(C13)), ((3 + (t * cos(C13 / 2))) * sin(C13)), (t * sin(C13 / 2)), t, -1, 1) → D13
  26. CurveCartesian(((3 + (t * cos(C14 / 2))) * cos(C14)), ((3 + (t * cos(C14 / 2))) * sin(C14)), (t * sin(C14 / 2)), t, -1, 1) → D14
  27. CurveCartesian(((3 + (t * cos(C15 / 2))) * cos(C15)), ((3 + (t * cos(C15 / 2))) * sin(C15)), (t * sin(C15 / 2)), t, -1, 1) → D15
  28. CurveCartesian(((3 + (t * cos(C16 / 2))) * cos(C16)), ((3 + (t * cos(C16 / 2))) * sin(C16)), (t * sin(C16 / 2)), t, -1, 1) → D16
  29. CurveCartesian(((3 + (t * cos(C17 / 2))) * cos(C17)), ((3 + (t * cos(C17 / 2))) * sin(C17)), (t * sin(C17 / 2)), t, -1, 1) → D17
  30. CurveCartesian(((3 + (t * cos(C18 / 2))) * cos(C18)), ((3 + (t * cos(C18 / 2))) * sin(C18)), (t * sin(C18 / 2)), t, -1, 1) → D18
  31. CurveCartesian(((3 + (t * cos(C19 / 2))) * cos(C19)), ((3 + (t * cos(C19 / 2))) * sin(C19)), (t * sin(C19 / 2)), t, -1, 1) → D19
  32. CurveCartesian(((3 + (t * cos(C20 / 2))) * cos(C20)), ((3 + (t * cos(C20 / 2))) * sin(C20)), (t * sin(C20 / 2)), t, -1, 1) → D20
  33. CurveCartesian(((3 + (t * cos(C21 / 2))) * cos(C21)), ((3 + (t * cos(C21 / 2))) * sin(C21)), (t * sin(C21 / 2)), t, -1, 1) → D21
  34. CurveCartesian(((3 + (t * cos(C22 / 2))) * cos(C22)), ((3 + (t * cos(C22 / 2))) * sin(C22)), (t * sin(C22 / 2)), t, -1, 1) → D22
  35. CurveCartesian(((3 + (t * cos(C23 / 2))) * cos(C23)), ((3 + (t * cos(C23 / 2))) * sin(C23)), (t * sin(C23 / 2)), t, -1, 1) → D23
  36. CurveCartesian(((3 + (t * cos(C24 / 2))) * cos(C24)), ((3 + (t * cos(C24 / 2))) * sin(C24)), (t * sin(C24 / 2)), t, -1, 1) → D24
  37. CurveCartesian(((3 + (t * cos(C25 / 2))) * cos(C25)), ((3 + (t * cos(C25 / 2))) * sin(C25)), (t * sin(C25 / 2)), t, -1, 1) → D25
  38. CurveCartesian(((3 + (t * cos(C26 / 2))) * cos(C26)), ((3 + (t * cos(C26 / 2))) * sin(C26)), (t * sin(C26 / 2)), t, -1, 1) → D26
  39. CurveCartesian(((3 + (t * cos(C27 / 2))) * cos(C27)), ((3 + (t * cos(C27 / 2))) * sin(C27)), (t * sin(C27 / 2)), t, -1, 1) → D27
  40. CurveCartesian(((3 + (t * cos(C28 / 2))) * cos(C28)), ((3 + (t * cos(C28 / 2))) * sin(C28)), (t * sin(C28 / 2)), t, -1, 1) → D28
  41. CurveCartesian(((3 + (t * cos(C29 / 2))) * cos(C29)), ((3 + (t * cos(C29 / 2))) * sin(C29)), (t * sin(C29 / 2)), t, -1, 1) → D29
  42. CurveCartesian(((3 + (t * cos(C30 / 2))) * cos(C30)), ((3 + (t * cos(C30 / 2))) * sin(C30)), (t * sin(C30 / 2)), t, -1, 1) → D30
  43. CurveCartesian(((3 + (t * cos(C31 / 2))) * cos(C31)), ((3 + (t * cos(C31 / 2))) * sin(C31)), (t * sin(C31 / 2)), t, -1, 1) → D31
  44. CurveCartesian(((3 + (t * cos(C32 / 2))) * cos(C32)), ((3 + (t * cos(C32 / 2))) * sin(C32)), (t * sin(C32 / 2)), t, -1, 1) → D32
  45. CurveCartesian(((3 + (t * cos(C33 / 2))) * cos(C33)), ((3 + (t * cos(C33 / 2))) * sin(C33)), (t * sin(C33 / 2)), t, -1, 1) → D33
  46. CurveCartesian(((3 + (t * cos(C34 / 2))) * cos(C34)), ((3 + (t * cos(C34 / 2))) * sin(C34)), (t * sin(C34 / 2)), t, -1, 1) → D34
  47. CurveCartesian(((3 + (t * cos(C35 / 2))) * cos(C35)), ((3 + (t * cos(C35 / 2))) * sin(C35)), (t * sin(C35 / 2)), t, -1, 1) → D35
  48. CurveCartesian(((3 + (t * cos(C36 / 2))) * cos(C36)), ((3 + (t * cos(C36 / 2))) * sin(C36)), (t * sin(C36 / 2)), t, -1, 1) → D36
  49. CurveCartesian(((3 + (t * cos(C37 / 2))) * cos(C37)), ((3 + (t * cos(C37 / 2))) * sin(C37)), (t * sin(C37 / 2)), t, -1, 1) → D37
  50. CurveCartesian(((3 + (t * cos(C38 / 2))) * cos(C38)), ((3 + (t * cos(C38 / 2))) * sin(C38)), (t * sin(C38 / 2)), t, -1, 1) → D38
  51. CurveCartesian(((3 + (t * cos(C39 / 2))) * cos(C39)), ((3 + (t * cos(C39 / 2))) * sin(C39)), (t * sin(C39 / 2)), t, -1, 1) → D39
  52. CurveCartesian(((3 + (t * cos(C40 / 2))) * cos(C40)), ((3 + (t * cos(C40 / 2))) * sin(C40)), (t * sin(C40 / 2)), t, -1, 1) → D40
  53. CurveCartesian(((3 + (t * cos(C41 / 2))) * cos(C41)), ((3 + (t * cos(C41 / 2))) * sin(C41)), (t * sin(C41 / 2)), t, -1, 1) → D41
  54. CurveCartesian(((3 + (t * cos(C42 / 2))) * cos(C42)), ((3 + (t * cos(C42 / 2))) * sin(C42)), (t * sin(C42 / 2)), t, -1, 1) → D42
  55. CurveCartesian(((3 + (t * cos(C43 / 2))) * cos(C43)), ((3 + (t * cos(C43 / 2))) * sin(C43)), (t * sin(C43 / 2)), t, -1, 1) → D43
  56. CurveCartesian(((3 + (t * cos(C44 / 2))) * cos(C44)), ((3 + (t * cos(C44 / 2))) * sin(C44)), (t * sin(C44 / 2)), t, -1, 1) → D44
  57. CurveCartesian(((3 + (t * cos(C45 / 2))) * cos(C45)), ((3 + (t * cos(C45 / 2))) * sin(C45)), (t * sin(C45 / 2)), t, -1, 1) → D45
  58. CurveCartesian(((3 + (t * cos(C46 / 2))) * cos(C46)), ((3 + (t * cos(C46 / 2))) * sin(C46)), (t * sin(C46 / 2)), t, -1, 1) → D46
  59. CurveCartesian(((3 + (t * cos(C47 / 2))) * cos(C47)), ((3 + (t * cos(C47 / 2))) * sin(C47)), (t * sin(C47 / 2)), t, -1, 1) → D47
  60. CurveCartesian(((3 + (t * cos(C48 / 2))) * cos(C48)), ((3 + (t * cos(C48 / 2))) * sin(C48)), (t * sin(C48 / 2)), t, -1, 1) → D48
  61. CurveCartesian(((3 + (t * cos(C49 / 2))) * cos(C49)), ((3 + (t * cos(C49 / 2))) * sin(C49)), (t * sin(C49 / 2)), t, -1, 1) → D49
  62. CurveCartesian(((3 + (t * cos(C50 / 2))) * cos(C50)), ((3 + (t * cos(C50 / 2))) * sin(C50)), (t * sin(C50 / 2)), t, -1, 1) → D50
  63. CurveCartesian(((3 + (t * cos(C51 / 2))) * cos(C51)), ((3 + (t * cos(C51 / 2))) * sin(C51)), (t * sin(C51 / 2)), t, -1, 1) → D51
  64. CurveCartesian(((3 + (t * cos(C52 / 2))) * cos(C52)), ((3 + (t * cos(C52 / 2))) * sin(C52)), (t * sin(C52 / 2)), t, -1, 1) → D52
  65. CurveCartesian(((3 + (t * cos(C53 / 2))) * cos(C53)), ((3 + (t * cos(C53 / 2))) * sin(C53)), (t * sin(C53 / 2)), t, -1, 1) → D53
  66. CurveCartesian(((3 + (t * cos(C54 / 2))) * cos(C54)), ((3 + (t * cos(C54 / 2))) * sin(C54)), (t * sin(C54 / 2)), t, -1, 1) → D54
  67. CurveCartesian(((3 + (t * cos(C55 / 2))) * cos(C55)), ((3 + (t * cos(C55 / 2))) * sin(C55)), (t * sin(C55 / 2)), t, -1, 1) → D55
  68. CurveCartesian(((3 + (t * cos(C56 / 2))) * cos(C56)), ((3 + (t * cos(C56 / 2))) * sin(C56)), (t * sin(C56 / 2)), t, -1, 1) → D56
  69. CurveCartesian(((3 + (t * cos(C57 / 2))) * cos(C57)), ((3 + (t * cos(C57 / 2))) * sin(C57)), (t * sin(C57 / 2)), t, -1, 1) → D57
  70. CurveCartesian(((3 + (t * cos(C58 / 2))) * cos(C58)), ((3 + (t * cos(C58 / 2))) * sin(C58)), (t * sin(C58 / 2)), t, -1, 1) → D58
  71. CurveCartesian(((3 + (t * cos(C59 / 2))) * cos(C59)), ((3 + (t * cos(C59 / 2))) * sin(C59)), (t * sin(C59 / 2)), t, -1, 1) → D59
  72. CurveCartesian(((3 + (t * cos(C60 / 2))) * cos(C60)), ((3 + (t * cos(C60 / 2))) * sin(C60)), (t * sin(C60 / 2)), t, -1, 1) → D60
  73. CurveCartesian(((3 + (t * cos(C61 / 2))) * cos(C61)), ((3 + (t * cos(C61 / 2))) * sin(C61)), (t * sin(C61 / 2)), t, -1, 1) → D61
  74. CurveCartesian(((3 + (t * cos(C62 / 2))) * cos(C62)), ((3 + (t * cos(C62 / 2))) * sin(C62)), (t * sin(C62 / 2)), t, -1, 1) → D62
  75. CurveCartesian(((3 + (t * cos(C63 / 2))) * cos(C63)), ((3 + (t * cos(C63 / 2))) * sin(C63)), (t * sin(C63 / 2)), t, -1, 1) → D63
  76. CurveCartesian(((3 + (t * cos(C64 / 2))) * cos(C64)), ((3 + (t * cos(C64 / 2))) * sin(C64)), (t * sin(C64 / 2)), t, -1, 1) → D64
  77. CurveCartesian(((3 + (t * cos(C65 / 2))) * cos(C65)), ((3 + (t * cos(C65 / 2))) * sin(C65)), (t * sin(C65 / 2)), t, -1, 1) → D65
  78. CurveCartesian(((3 + (t * cos(C66 / 2))) * cos(C66)), ((3 + (t * cos(C66 / 2))) * sin(C66)), (t * sin(C66 / 2)), t, -1, 1) → D66
  79. CurveCartesian(((3 + (t * cos(C67 / 2))) * cos(C67)), ((3 + (t * cos(C67 / 2))) * sin(C67)), (t * sin(C67 / 2)), t, -1, 1) → D67
  80. CurveCartesian(((3 + (t * cos(C68 / 2))) * cos(C68)), ((3 + (t * cos(C68 / 2))) * sin(C68)), (t * sin(C68 / 2)), t, -1, 1) → D68
  81. CurveCartesian(((3 + (t * cos(C69 / 2))) * cos(C69)), ((3 + (t * cos(C69 / 2))) * sin(C69)), (t * sin(C69 / 2)), t, -1, 1) → D69
  82. CurveCartesian(((3 + (t * cos(C70 / 2))) * cos(C70)), ((3 + (t * cos(C70 / 2))) * sin(C70)), (t * sin(C70 / 2)), t, -1, 1) → D70
  83. CurveCartesian(((3 + (t * cos(C71 / 2))) * cos(C71)), ((3 + (t * cos(C71 / 2))) * sin(C71)), (t * sin(C71 / 2)), t, -1, 1) → D71
  84. CurveCartesian(((3 + (t * cos(C72 / 2))) * cos(C72)), ((3 + (t * cos(C72 / 2))) * sin(C72)), (t * sin(C72 / 2)), t, -1, 1) → D72