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Knot on a Torus

Knot on a Torus
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2020-07-06 19:48
FunctionTrigonometryCoordinate GeometrySolid GeometryParametric Equation
This example demonstrates how to draw a knot on a torus surface. It involves core mathematical concepts such as surfaces, line segments, distributions, circumcircle, trigonometry, and functions. The construction uses GeoGebra commands/objects like CurveCartesian (Cartesian curve), Segment (line segment), Sequence (sequence), Element (element), Circle (circle), Polygon (polygon), Prism (prism), and Surface (surface). Suitable for high school and above students to understand spatial geometry and topological structures.

GeoGebra Commands & Parameters

  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