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Tangent at a General Point of Intersection of a Torus

Tangent at a General Point of Intersection of a Torus
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2020-04-09 02:00
CircleSolid GeometryTransformationGeometric ConstructionLocus
This example demonstrates the geometric construction of a tangent line at a general intersection point on a torus. It involves core concepts of solid geometry, including the torus surface, tangent planes, and local linearization at intersection points. The construction uses GeoGebra commands such as Line, Point, Circle, OrthogonalLine, Midpoint, Intersect, Mirror, Segment, Center, Locus, CircleArc, and Angle, along with objects like O', A, f, B, c, etc. Through operations like intersecting, drawing perpendiculars, finding midpoints, and reflecting, the tangent direction at the intersection point is determined. This is suitable for high school or undergraduate courses in solid geometry and analytic geometry, helping students understand tangent construction on curved surfaces and gain proficiency in interactive 3D GeoGebra tools.

GeoGebra Commands & Parameters

  1. Line(O', A) → f
  2. Point(f) → B
  3. Circle(O', B) → c
  4. Point(f) → C
  5. Circle(O', C) → d
  6. OrthogonalLine(O', f) → g
  7. Midpoint(B, C) → D
  8. OrthogonalLine(D, f) → h
  9. Point(g) → E
  10. OrthogonalLine(E, g) → i
  11. Intersect(h, i) → F
  12. Circle(F, Segment[D, C]) → e
  13. Mirror(e, g) → e'
  14. Mirror(h, g) → h'
  15. Intersect(e', h', 2) → G
  16. Intersect(e, h, 2) → H
  17. Segment(G, H) → j
  18. Intersect(e', h', 1) → I
  19. Intersect(e, h, 1) → J
  20. Segment(I, J) → k
  21. Line(O', K) → l
  22. OrthogonalLine(K, l) → n_1
  23. Line(I, J) → n_2
  24. Line(G, H) → p
  25. Point(l) → L
  26. OrthogonalLine(L, l) → q
  27. OrthogonalLine(K, f) → r
  28. OrthogonalLine(L, f) → s
  29. Intersect(r, n_2) → M
  30. Intersect(s, p) → N
  31. Segment(M, N) → t
  32. Point(t) → P_2
  33. OrthogonalLine(P_2, g) → a
  34. Intersect(e', a) → Q, P_0''p
  35. Intersect(e, a) → S, T
  36. Intersect(a, g) → U
  37. Circle(O', Segment[U, Q]) → c_1
  38. Circle(O', Segment[U, P_0''p]) → d_1
  39. OrthogonalLine(P_2, i) → b
  40. Intersect(l, b) → V
  41. OrthogonalLine(V, l) → f_1
  42. Center(e') → I_1
  43. Intersect(d_1, f_1) → P', Z
  44. Intersect(c_1, f_1) → A_1, B_1
  45. Locus(P', P_2) → ponthalmaz1
  46. Locus(A_1, P_2) → ponthalmaz2
  47. Locus(B_1, P_2) → ponthalmaz3
  48. Locus(Z, P_2) → ponthalmaz4
  49. OrthogonalLine(Z, f) → g_1
  50. OrthogonalLine(B_1, f) → h_1
  51. OrthogonalLine(A_1, f) → i_1
  52. OrthogonalLine(P', f) → j_1
  53. Intersect(g_1, a) → C_1
  54. Intersect(h_1, a) → D_1
  55. Segment(P_0''p, S) → k_2
  56. Intersect(i_1, a) → E_1
  57. Intersect(j_1, a) → P''
  58. Locus(P'', P_2) → ponthalmaz5
  59. Locus(E_1, P_2) → ponthalmaz6
  60. Locus(D_1, P_2) → ponthalmaz7
  61. Locus(C_1, P_2) → ponthalmaz8
  62. Intersect(d_1, f) → P_0', H_1
  63. Segment(O', P_0') → m
  64. Segment(P', O') → n
  65. Segment(P_0''p, P_0') → p_1
  66. Intersect(e', k_2, 1) → P_0''
  67. Segment(I_1, P_0''p) → k_1
  68. OrthogonalLine(P_0''p, k_1) → l_1
  69. Intersect(n_2, l_1) → J_1
  70. Segment(P_0''p, J_1) → m_1
  71. OrthogonalLine(J_1, f) → n_3
  72. Intersect(f, n_3) → K_1
  73. Segment(O', K_1) → l_2
  74. Segment(J_1, K_1) → n_0
  75. Line(O', P') → q_1
  76. Circle(O', K_1) → e_1
  77. Intersect(e_1, q_1) → L_1, M_1
  78. CircleArc(O', M_1, K_1) → r_1
  79. Segment(P', M_1) → t_1
  80. OrthogonalLine(M_1, q_1) → s_1
  81. Intersect(n_1, s_1) → N_1
  82. Angle(N_1, M_1, O') → α
  83. Segment(M_1, N_1) → n_P
  84. Line(N_1, P') → a_1
  85. Mirror(N_1, P') → O_1
  86. Segment(O_1, N_1) → t_P'
  87. OrthogonalLine(N_1, n_2) → f_2
  88. Intersect(n_2, f_2) → P_1
  89. Segment(N_1, P_1) → g_2
  90. Line(P_1, P'') → h_2
  91. Mirror(P_1, P'') → Q_1
  92. Segment(Q_1, P_1) → t_P''
  93. Segment(O', M_1) → j_2

AI Drawing Prompt Reference

Draw a 3D torus with a central axis line. Pick two distinct points on this line and create circles centered at the torus center passing through each point. Construct a perpendicular line through the center to the axis. Use midpoints and reflections to build auxiliary circles intersecting the torus surface. Determine a tangent direction at a general intersection point on the torus. Then create a segment along this tangent and slide it along the torus, generating multiple locus curves that trace how the tangent point moves as the segment shifts.
The prompt above was compiled from the original drawing commands and is for reference only.