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托尔里斯曲线上的舞动立方体II

托尔里斯曲线上的舞动立方体II
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2018-10-22 18:47
立体几何变换参数方程
这个范例展示了立方体在托尔里斯曲线上的动态变换。涉及的核心数学概念包括几何变换和参数方程。通过使用GeoGebra命令如Point, Cube, CurveCartesian, Rotate等，可以构造出立方体在曲线上的运动轨迹。适合高中及以上学生或教师用于教学和学习几何变换和参数方程。

GeoGebra 命令及参数

  1. Point(Circle[B, Distance[A, B], Segment[A, B]]) → C
  2. Cube(A, B, C) → a, D, faceCDHG, faceEFGH, edgeAD, edgeAB, edgeBC, edgeCD, edgeAE, edgeDH, edgeEH, edgeEF, E, edgeBF, edgeFG, edgeCG, edgeGH, F, G, H, faceABCD, faceADHE, faceABFE, faceBCGF
  3. CurveCartesian(((2 + cos((((q * i) * 2) * π))) * cos((((p * i) * 2) * π))), ((2 + cos((((q * i) * 2) * π))) * sin((((p * i) * 2) * π))), sin((((q * i) * 2) * π)), i, 0, 1) → knot
  4. Rotate(Translate[a, Vector[(0, 0, 0), knot(0 / n)]], t + ((2 * π) * 0 / n), knot(0 / n)) → A0
  5. Rotate(Translate[a, Vector[(0, 0, 0), knot(1 / n)]], t + ((2 * π) * 1 / n), knot(1 / n)) → A1
  6. Rotate(Translate[a, Vector[(0, 0, 0), knot(2 / n)]], t + ((2 * π) * 2 / n), knot(2 / n)) → A2
  7. Rotate(Translate[a, Vector[(0, 0, 0), knot(3 / n)]], t + ((2 * π) * 3 / n), knot(3 / n)) → A3
  8. Rotate(Translate[a, Vector[(0, 0, 0), knot(4 / n)]], t + ((2 * π) * 4 / n), knot(4 / n)) → A4
  9. Rotate(Translate[a, Vector[(0, 0, 0), knot(5 / n)]], t + ((2 * π) * 5 / n), knot(5 / n)) → A5
  10. Rotate(Translate[a, Vector[(0, 0, 0), knot(6 / n)]], t + ((2 * π) * 6 / n), knot(6 / n)) → A6
  11. Rotate(Translate[a, Vector[(0, 0, 0), knot(7 / n)]], t + ((2 * π) * 7 / n), knot(7 / n)) → A7
  12. Rotate(Translate[a, Vector[(0, 0, 0), knot(8 / n)]], t + ((2 * π) * 8 / n), knot(8 / n)) → A8
  13. Rotate(Translate[a, Vector[(0, 0, 0), knot(9 / n)]], t + ((2 * π) * 9 / n), knot(9 / n)) → A9
  14. Rotate(Translate[a, Vector[(0, 0, 0), knot(10 / n)]], t + ((2 * π) * 10 / n), knot(10 / n)) → A10
  15. Rotate(Translate[a, Vector[(0, 0, 0), knot(11 / n)]], t + ((2 * π) * 11 / n), knot(11 / n)) → A11
  16. Rotate(Translate[a, Vector[(0, 0, 0), knot(12 / n)]], t + ((2 * π) * 12 / n), knot(12 / n)) → A12
  17. Rotate(Translate[a, Vector[(0, 0, 0), knot(13 / n)]], t + ((2 * π) * 13 / n), knot(13 / n)) → A13
  18. Rotate(Translate[a, Vector[(0, 0, 0), knot(14 / n)]], t + ((2 * π) * 14 / n), knot(14 / n)) → A14
  19. Rotate(Translate[a, Vector[(0, 0, 0), knot(15 / n)]], t + ((2 * π) * 15 / n), knot(15 / n)) → A15
  20. Rotate(Translate[a, Vector[(0, 0, 0), knot(16 / n)]], t + ((2 * π) * 16 / n), knot(16 / n)) → A16
  21. Rotate(Translate[a, Vector[(0, 0, 0), knot(17 / n)]], t + ((2 * π) * 17 / n), knot(17 / n)) → A17
  22. Rotate(Translate[a, Vector[(0, 0, 0), knot(18 / n)]], t + ((2 * π) * 18 / n), knot(18 / n)) → A18
  23. Rotate(Translate[a, Vector[(0, 0, 0), knot(19 / n)]], t + ((2 * π) * 19 / n), knot(19 / n)) → A19
  24. Rotate(Translate[a, Vector[(0, 0, 0), knot(20 / n)]], t + ((2 * π) * 20 / n), knot(20 / n)) → A20
  25. Rotate(Translate[a, Vector[(0, 0, 0), knot(21 / n)]], t + ((2 * π) * 21 / n), knot(21 / n)) → A21
  26. Rotate(Translate[a, Vector[(0, 0, 0), knot(22 / n)]], t + ((2 * π) * 22 / n), knot(22 / n)) → A22
  27. Rotate(Translate[a, Vector[(0, 0, 0), knot(23 / n)]], t + ((2 * π) * 23 / n), knot(23 / n)) → A23
  28. Rotate(Translate[a, Vector[(0, 0, 0), knot(24 / n)]], t + ((2 * π) * 24 / n), knot(24 / n)) → A24
  29. Rotate(Translate[a, Vector[(0, 0, 0), knot(25 / n)]], t + ((2 * π) * 25 / n), knot(25 / n)) → A25
  30. Rotate(Translate[a, Vector[(0, 0, 0), knot(26 / n)]], t + ((2 * π) * 26 / n), knot(26 / n)) → A26
  31. Rotate(Translate[a, Vector[(0, 0, 0), knot(27 / n)]], t + ((2 * π) * 27 / n), knot(27 / n)) → A27
  32. Rotate(Translate[a, Vector[(0, 0, 0), knot(28 / n)]], t + ((2 * π) * 28 / n), knot(28 / n)) → A28
  33. Rotate(Translate[a, Vector[(0, 0, 0), knot(29 / n)]], t + ((2 * π) * 29 / n), knot(29 / n)) → A29
  34. Rotate(Translate[a, Vector[(0, 0, 0), knot(30 / n)]], t + ((2 * π) * 30 / n), knot(30 / n)) → A30
  35. Rotate(Translate[a, Vector[(0, 0, 0), knot(31 / n)]], t + ((2 * π) * 31 / n), knot(31 / n)) → A31
  36. Rotate(Translate[a, Vector[(0, 0, 0), knot(32 / n)]], t + ((2 * π) * 32 / n), knot(32 / n)) → A32
  37. Rotate(Translate[a, Vector[(0, 0, 0), knot(33 / n)]], t + ((2 * π) * 33 / n), knot(33 / n)) → A33
  38. Rotate(Translate[a, Vector[(0, 0, 0), knot(34 / n)]], t + ((2 * π) * 34 / n), knot(34 / n)) → A34
  39. Rotate(Translate[a, Vector[(0, 0, 0), knot(35 / n)]], t + ((2 * π) * 35 / n), knot(35 / n)) → A35
  40. Rotate(Translate[a, Vector[(0, 0, 0), knot(36 / n)]], t + ((2 * π) * 36 / n), knot(36 / n)) → A36
  41. Rotate(Translate[a, Vector[(0, 0, 0), knot(37 / n)]], t + ((2 * π) * 37 / n), knot(37 / n)) → A37
  42. Rotate(Translate[a, Vector[(0, 0, 0), knot(38 / n)]], t + ((2 * π) * 38 / n), knot(38 / n)) → A38
  43. Rotate(Translate[a, Vector[(0, 0, 0), knot(39 / n)]], t + ((2 * π) * 39 / n), knot(39 / n)) → A39
  44. Rotate(Translate[a, Vector[(0, 0, 0), knot(40 / n)]], t + ((2 * π) * 40 / n), knot(40 / n)) → A40
  45. Rotate(Translate[a, Vector[(0, 0, 0), knot(41 / n)]], t + ((2 * π) * 41 / n), knot(41 / n)) → A41
  46. Rotate(Translate[a, Vector[(0, 0, 0), knot(42 / n)]], t + ((2 * π) * 42 / n), knot(42 / n)) → A42
  47. Rotate(Translate[a, Vector[(0, 0, 0), knot(43 / n)]], t + ((2 * π) * 43 / n), knot(43 / n)) → A43
  48. Rotate(Translate[a, Vector[(0, 0, 0), knot(44 / n)]], t + ((2 * π) * 44 / n), knot(44 / n)) → A44
  49. Rotate(Translate[a, Vector[(0, 0, 0), knot(45 / n)]], t + ((2 * π) * 45 / n), knot(45 / n)) → A45
  50. Rotate(Translate[a, Vector[(0, 0, 0), knot(46 / n)]], t + ((2 * π) * 46 / n), knot(46 / n)) → A46
  51. Rotate(Translate[a, Vector[(0, 0, 0), knot(47 / n)]], t + ((2 * π) * 47 / n), knot(47 / n)) → A47
  52. Rotate(Translate[a, Vector[(0, 0, 0), knot(48 / n)]], t + ((2 * π) * 48 / n), knot(48 / n)) → A48
  53. Rotate(Translate[a, Vector[(0, 0, 0), knot(49 / n)]], t + ((2 * π) * 49 / n), knot(49 / n)) → A49
  54. Rotate(Translate[a, Vector[(0, 0, 0), knot(50 / n)]], t + ((2 * π) * 50 / n), knot(50 / n)) → A50
  55. Rotate(Translate[a, Vector[(0, 0, 0), knot(51 / n)]], t + ((2 * π) * 51 / n), knot(51 / n)) → A51
  56. Rotate(Translate[a, Vector[(0, 0, 0), knot(52 / n)]], t + ((2 * π) * 52 / n), knot(52 / n)) → A52
  57. Rotate(Translate[a, Vector[(0, 0, 0), knot(53 / n)]], t + ((2 * π) * 53 / n), knot(53 / n)) → A53
  58. Rotate(Translate[a, Vector[(0, 0, 0), knot(54 / n)]], t + ((2 * π) * 54 / n), knot(54 / n)) → A54
  59. Rotate(Translate[a, Vector[(0, 0, 0), knot(55 / n)]], t + ((2 * π) * 55 / n), knot(55 / n)) → A55
  60. Rotate(Translate[a, Vector[(0, 0, 0), knot(56 / n)]], t + ((2 * π) * 56 / n), knot(56 / n)) → A56
  61. Rotate(Translate[a, Vector[(0, 0, 0), knot(57 / n)]], t + ((2 * π) * 57 / n), knot(57 / n)) → A57
  62. Rotate(Translate[a, Vector[(0, 0, 0), knot(58 / n)]], t + ((2 * π) * 58 / n), knot(58 / n)) → A58
  63. Rotate(Translate[a, Vector[(0, 0, 0), knot(59 / n)]], t + ((2 * π) * 59 / n), knot(59 / n)) → A59
  64. Rotate(Translate[a, Vector[(0, 0, 0), knot(60 / n)]], t + ((2 * π) * 60 / n), knot(60 / n)) → A60
  65. Rotate(Translate[a, Vector[(0, 0, 0), knot(61 / n)]], t + ((2 * π) * 61 / n), knot(61 / n)) → A61
  66. Rotate(Translate[a, Vector[(0, 0, 0), knot(62 / n)]], t + ((2 * π) * 62 / n), knot(62 / n)) → A62
  67. Rotate(Translate[a, Vector[(0, 0, 0), knot(63 / n)]], t + ((2 * π) * 63 / n), knot(63 / n)) → A63
  68. Rotate(Translate[a, Vector[(0, 0, 0), knot(64 / n)]], t + ((2 * π) * 64 / n), knot(64 / n)) → A64
  69. Rotate(Translate[a, Vector[(0, 0, 0), knot(65 / n)]], t + ((2 * π) * 65 / n), knot(65 / n)) → A65
  70. Rotate(Translate[a, Vector[(0, 0, 0), knot(66 / n)]], t + ((2 * π) * 66 / n), knot(66 / n)) → A66
  71. Rotate(Translate[a, Vector[(0, 0, 0), knot(67 / n)]], t + ((2 * π) * 67 / n), knot(67 / n)) → A67
  72. Rotate(Translate[a, Vector[(0, 0, 0), knot(68 / n)]], t + ((2 * π) * 68 / n), knot(68 / n)) → A68
  73. Rotate(Translate[a, Vector[(0, 0, 0), knot(69 / n)]], t + ((2 * π) * 69 / n), knot(69 / n)) → A69
  74. Rotate(Translate[a, Vector[(0, 0, 0), knot(70 / n)]], t + ((2 * π) * 70 / n), knot(70 / n)) → A70
  75. Rotate(Translate[a, Vector[(0, 0, 0), knot(71 / n)]], t + ((2 * π) * 71 / n), knot(71 / n)) → A71
  76. Rotate(Translate[a, Vector[(0, 0, 0), knot(72 / n)]], t + ((2 * π) * 72 / n), knot(72 / n)) → A72
  77. Rotate(Translate[a, Vector[(0, 0, 0), knot(73 / n)]], t + ((2 * π) * 73 / n), knot(73 / n)) → A73
  78. Rotate(Translate[a, Vector[(0, 0, 0), knot(74 / n)]], t + ((2 * π) * 74 / n), knot(74 / n)) → A74
  79. Rotate(Translate[a, Vector[(0, 0, 0), knot(75 / n)]], t + ((2 * π) * 75 / n), knot(75 / n)) → A75
  80. Rotate(Translate[a, Vector[(0, 0, 0), knot(76 / n)]], t + ((2 * π) * 76 / n), knot(76 / n)) → A76
  81. Rotate(Translate[a, Vector[(0, 0, 0), knot(77 / n)]], t + ((2 * π) * 77 / n), knot(77 / n)) → A77
  82. Rotate(Translate[a, Vector[(0, 0, 0), knot(78 / n)]], t + ((2 * π) * 78 / n), knot(78 / n)) → A78
  83. Rotate(Translate[a, Vector[(0, 0, 0), knot(79 / n)]], t + ((2 * π) * 79 / n), knot(79 / n)) → A79