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环面上的结

环面上的结
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2020-07-06 19:48
函数三角函数解析几何立体几何参数方程
①本范例演示了如何在环面上绘制一个结；②涉及的核心数学概念包括曲面、线段、分布、外接圆、三角函数和函数；③作图要点——从下方『构造摘要』提炼用到了 CurveCartesian（笛卡尔曲线）、Segment（线段）、Sequence（序列）、Element（元素）、Circle（圆）、Polygon（多边形）、Prism（棱柱）、Surface（曲面）等 GeoGebra 命令/对象；④适合高中及以上数学学习者，用于理解空间几何与拓扑结构。

GeoGebra 命令及参数

  1. CurveCartesian((cos(s) * (a + (ra * cos(((2 * m) * s / 6))))), (sin(s) * (a + (ra * cos(((2 * m) * s / 6))))), (ra * sin(((2 * m) * s / 6))), s, 0, 6.28319) → C_1
  2. CurveCartesian((cos(s) * (a + (ra * cos(((2 * m) * s / 6) + (2 * π / 3))))), (sin(s) * (a + (ra * cos(((2 * m) * s / 6) + (2 * π / 3))))), (ra * sin(((2 * m) * s / 6) + (2 * π / 3))), s, 0, 6.28319) → C_2
  3. CurveCartesian((cos(s) * (a + (ra * cos(((2 * m) * s / 6) + (4 * π / 3))))), (sin(s) * (a + (ra * cos(((2 * m) * s / 6) + (4 * π / 3))))), (ra * sin(((2 * m) * s / 6) + (4 * π / 3))), s, 0, 6.28319) → C_3
  4. Segment(A, C) → b
  5. Segment(B, C) → c
  6. Segment(B, A) → d
  7. Sequence(((cos(s) * (a + (ra * cos(((2 * m) * s / 6) + (2 * π / 3))))), (sin(s) * (a + (ra * cos(((2 * m) * s / 6) + (2 * π / 3))))), (ra * sin(((2 * m) * s / 6) + (2 * π / 3)))), s, 0, 6.28319, (2 * π / n)) → l2
  8. Sequence(((cos(s) * (a + (ra * cos(((2 * m) * s / 6))))), (sin(s) * (a + (ra * cos(((2 * m) * s / 6))))), (ra * sin(((2 * m) * s / 6)))), s, 0, 6.28319, (2 * π / n)) → l1
  9. Sequence(((cos(s) * (a + (ra * cos(((2 * m) * s / 6) + (4 * π / 3))))), (sin(s) * (a + (ra * cos(((2 * m) * s / 6) + (4 * π / 3))))), (ra * sin(((2 * m) * s / 6) + (4 * π / 3)))), s, 0, 6.28319, (2 * π / n)) → l3
  10. Sequence(Polygon[Element[l1, i], Element[l2, i], Element[l3, i]], i, 1, Length[l1]) → l4
  11. Sequence(Prism[Element[l4, i], 1], i, 1, Length[l4]) → l5
  12. Element(l5, A1) → B1
  13. Element(l5, A2) → B2
  14. Element(l5, A3) → B3
  15. Element(l5, A4) → B4
  16. Element(l5, A5) → B5
  17. Element(l5, A6) → B6
  18. Element(l5, A7) → B7
  19. Element(l5, A8) → B8
  20. Element(l5, A9) → B9
  21. Element(l5, A10) → B10
  22. Element(l5, A11) → B11
  23. Element(l5, A12) → B12
  24. Element(l5, A13) → B13
  25. Element(l5, A14) → B14
  26. Element(l5, A15) → B15
  27. Element(l5, A16) → B16
  28. Element(l5, A17) → B17
  29. Element(l5, A18) → B18
  30. Element(l5, A19) → B19
  31. Element(l5, A20) → B20
  32. Element(l5, A21) → B21
  33. Element(l5, A22) → B22
  34. Element(l5, A23) → B23
  35. Element(l5, A24) → B24
  36. Element(l5, A25) → B25
  37. Element(l5, A26) → B26
  38. Element(l5, A27) → B27
  39. Element(l5, A28) → B28
  40. Element(l5, A29) → B29
  41. Element(l5, A30) → B30
  42. Element(l5, A31) → B31
  43. Element(l5, A32) → B32
  44. Element(l5, A33) → B33
  45. Element(l5, A34) → B34
  46. Element(l5, A35) → B35
  47. Element(l5, A36) → B36
  48. Element(l5, A37) → B37
  49. Element(l5, A38) → B38
  50. Element(l5, A39) → B39
  51. Element(l5, A40) → B40
  52. Element(l5, A41) → B41
  53. Element(l5, A42) → B42
  54. Element(l5, A43) → B43
  55. Element(l5, A44) → B44
  56. Element(l5, A45) → B45
  57. Element(l5, A46) → B46
  58. Element(l5, A47) → B47
  59. Element(l5, A48) → B48
  60. Element(l5, A49) → B49
  61. Element(l5, A50) → B50
  62. Element(l5, A51) → B51
  63. Element(l5, A52) → B52
  64. Element(l5, A53) → B53
  65. Element(l5, A54) → B54
  66. Element(l5, A55) → B55
  67. Element(l5, A56) → B56
  68. Element(l5, A57) → B57
  69. Element(l5, A58) → B58
  70. Element(l5, A59) → B59
  71. Element(l5, A60) → B60
  72. Element(l5, A61) → B61
  73. Element(l5, A62) → B62
  74. Element(l5, A63) → B63
  75. Element(l5, A64) → B64
  76. Element(l5, A65) → B65
  77. Element(l5, A66) → B66
  78. Element(l5, A67) → B67
  79. Element(l5, A68) → B68
  80. Element(l5, A69) → B69
  81. Element(l5, A70) → B70
  82. Element(l5, A71) → B71
  83. Element(l5, A72) → B72
  84. Element(l5, A73) → B73
  85. Element(l5, A74) → B74
  86. Element(l5, A75) → B75
  87. Element(l5, A76) → B76
  88. Element(l5, A77) → B77
  89. Element(l5, A78) → B78
  90. Element(l5, A79) → B79
  91. Element(l5, A80) → B80
  92. Element(l5, A81) → B81
  93. Element(l5, A82) → B82
  94. Element(l5, A83) → B83
  95. Element(l5, A84) → B84
  96. Element(l5, A85) → B85
  97. Element(l5, A86) → B86
  98. Element(l5, A87) → B87
  99. Element(l5, A88) → B88
  100. Element(l5, A89) → B89
  101. Element(l5, A90) → B90
  102. Element(l5, A91) → B91
  103. Element(l5, A92) → B92
  104. Element(l5, A93) → B93
  105. Element(l5, A94) → B94
  106. Element(l5, A95) → B95
  107. Element(l5, A96) → B96
  108. Element(l5, A97) → B97
  109. Element(l5, A98) → B98
  110. Element(l5, A99) → B99
  111. Element(l5, A100) → B100
  112. Circle(A, C, B) → e
  113. Polygon(A, B, C) → t1, c_1, a_1, b_1
  114. Prism(t1, 0.5) → g, D, arête3, arête4, arête5, arête6, E, F, face1, face2, face3, face4, arête1, arête2
  115. Prism(t1, -0.5) → h, G, arêteCI, arêteGH, arêteHI, arêteGI, H, I, faceABHG, faceBCIH, faceACIG, faceGHI, arêteAG, arêteBH
  116. Circle(K, J, L) → k
  117. Surface(k, (2 * π), zAxis) → i