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Intersection of Two Cylinders: Constructing Spatial Curves via Mirror and Orthog

Intersection of Two Cylinders: Constructing Spatial Curves via Mirror and Orthog
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2021-03-05 03:07
CircleSolid GeometryTransformationGeometric Construction
This example demonstrates how the intersection line (the common curve) of two cylinders in 3D space can be constructed and visualized precisely. It focuses on solid-geometry ideas such as the intersection of two surfaces, the geometric meaning of a “set intersection,” and the use of symmetry and orthogonality to obtain reliable auxiliary constructions. The construction leverages GeoGebra commands like Intersect to compute intersection objects, Circle to define circular cross-sections or reference curves, Mirror for symmetric placement, and OrthogonalLine to enforce perpendicular constraints. It also uses Point, Line, Segment, Ray, and Midpoint for building the skeleton of the construction, along with Rotate when a rotated configuration is needed; Locus and Semicircle help organize curve relationships and boundary pieces. Suitable for advanced middle school, high school, and early university solid geometry classes, as well as for interactive math visualization and modeling practice.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Circle(A, 1) → c
  3. Intersect(c, xAxis) → B, C
  4. Mirror(C, B) → C'
  5. OrthogonalLine(C', xAxis) → f
  6. OrthogonalLine(C, xAxis) → g
  7. OrthogonalLine(B, xAxis) → h_{2}
  8. Point(h_{2}) → D
  9. OrthogonalLine(D, h_{2}) → i
  10. Intersect(f, i) → E
  11. Circle(D, E) → d
  12. Intersect(i, g) → F
  13. Rotate(F, α, D) → F'
  14. Ray(D, F') → j
  15. OrthogonalLine(F', j) → k
  16. OrthogonalLine(D, j) → l
  17. Intersect(d, l, 2) → G
  18. OrthogonalLine(G, i) → m
  19. Intersect(c, yAxis) → H, I
  20. Intersect(m, xAxis) → J
  21. Midpoint(D, F') → K
  22. OrthogonalLine(K, j) → n
  23. Intersect(d, n) → L, M
  24. OrthogonalLine(M, i) → p
  25. OrthogonalLine(L, i) → q
  26. OrthogonalLine(I, yAxis) → r
  27. OrthogonalLine(H, yAxis) → s
  28. OrthogonalLine(F', i) → t
  29. Intersect(t, xAxis) → N
  30. Intersect(p, r) → O
  31. Intersect(q, s) → P
  32. OrthogonalLine(K, i) → f_1
  33. OrthogonalLine(D, i) → g_1
  34. Intersect(d, l, 1) → U
  35. OrthogonalLine(U, i) → h_1
  36. Intersect(xAxis, h_1) → V
  37. Intersect(q, r) → W
  38. Intersect(p, s) → Z
  39. Segment(D, F') → i_1
  40. Point(i_1) → S
  41. Line(S, n) → j_1
  42. Intersect(d, j_1) → T, A_1
  43. Line(A_1, yAxis) → r_1
  44. Semicircle(F', D) → c_1
  45. Intersect(c_1, j_1, 1) → F_1
  46. Segment(S, F_1) → h
  47. Circle(A, h) → d_1
  48. Intersect(d_1, yAxis) → G_1, H_1
  49. OrthogonalLine(H_1, yAxis) → s_1
  50. OrthogonalLine(G_1, yAxis) → t_1
  51. Intersect(r_1, s_1) → I_1
  52. Intersect(r_1, t_1) → J_1
  53. Locus(I_1, S) → lieu1
  54. Locus(J_1, S) → lieu2
  55. OrthogonalLine(T, i) → a_1
  56. Intersect(a_1, s_1) → K_1
  57. Intersect(t_1, a_1) → L_1
  58. Locus(K_1, S) → lieu3
  59. Locus(L_1, S) → lieu4
  60. Segment(A, H_1) → a
  61. Point(l) → Q
  62. Point(l) → R
  63. OrthogonalLine(R, l) → b
  64. OrthogonalLine(Q, l) → k_1
  65. Intersect(k, k_1) → B_1
  66. Intersect(k, b) → C_1
  67. Segment(B_1, C_1) → l_1
  68. Segment(Q, R) → m_1
  69. Intersect(c_1, l_1, 1) → D_1
  70. Intersect(d, i) → M_1, N_1
  71. OrthogonalLine(M_1, i) → n_1
  72. OrthogonalLine(N_1, i) → q_1
  73. Point(n_1) → P_1
  74. Point(n_1) → Q_1
  75. OrthogonalLine(P_1, n_1) → b_1
  76. OrthogonalLine(Q_1, n_1) → e_1
  77. Segment(Q_1, P_1) → f_2
  78. Line(J, A) → i_2
  79. Segment(S, A_1) → j_2
  80. Intersect(n, k_1) → W_1
  81. Ray(W_1, M) → k_2
  82. Intersect(g, b_1) → Z_1
  83. Intersect(g, e_1) → A_2
  84. Segment(A_2, Z_1) → g_2
  85. OrthogonalLine(T, i_2) → l_2
  86. Segment(F_1, T) → m_2
  87. Vague(Q_1, A_2) → p_1, c_2
  88. Vague(Q, B_1) → d_2, e_2
  89. Vague(R, C_1) → p_2, q_2
  90. Vague(P_1, Z_1) → r_2, s_2

AI Drawing Prompt Reference

Draw the intersection curve of two cylinders in 3D space. Use symmetry and orthogonality to assist in locating the curve. The figure includes circles, points, lines, rays, and other elements. First, define circles on the coordinate axes. Then, use symmetry to get symmetric points. Next, construct auxiliary lines using orthogonal lines. Finally, determine the curve's position using these auxiliary lines. Add some auxiliary points, rays, and semicircles to complete the figure, making the construction process clear.
The prompt above was compiled from the original drawing commands and is for reference only.