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Geometric Construction of Cylinder–Cone Intersections

Geometric Construction of Cylinder–Cone Intersections
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2020-09-12 14:53
CircleSolid GeometryTransformationGeometric ConstructionSlider Animation
This example demonstrates how the intersection curve of a cylinder and a cone can be reconstructed and visualized while dragging a control (parameter) point. It provides an intuitive way to observe how the spatial intersection changes with configuration. Key concepts include solid geometry of cylinders and cones, intersections between lines/segments in 3D construction workflows, circle-based auxiliary constructions, and tangent constraints. The construction relies on GeoGebra commands listed in the summary: Line and Ray to define supporting directions, Intersect and Segment to compute intersection points, Circle and Tangent to enforce tangency conditions, OrthogonalLine to create perpendicular helpers, and Mirror to use symmetry for building corresponding parts. Using Points and multiple labeled objects (including mirrored/primed counterparts) the model assembles the full intersection geometry. Suitable for high-school to early university learners studying 3D geometry, geometric construction, and visualization for math modeling.

GeoGebra Commands & Parameters

  1. OrthogonalLine(A, xAxis) → f
  2. Ray(A, B) → g
  3. Intersect(g, xAxis) → C
  4. Segment(C, A) → h
  5. Mirror(C, f) → C'
  6. Mirror(A, f) → A'
  7. Segment(C', A') → h'
  8. Point(f) → D
  9. Circle(D, E) → c
  10. Line(C, f) → i
  11. Point(f) → F
  12. Line(F, xAxis) → j
  13. Intersect(j, i) → G
  14. Circle(F, G) → d
  15. Tangent(f, c) → k, l
  16. Tangent(j, d) → m, n
  17. Point(j) → H
  18. Line(H, yAxis) → p
  19. Intersect(p, n) → I
  20. Circle(H, I) → e
  21. Tangent(p, e) → q, r
  22. Intersect(q, xAxis) → J
  23. Intersect(r, xAxis) → K
  24. Line(A, xAxis) → s
  25. Intersect(s, p) → L
  26. Segment(J, L) → t
  27. Segment(L, K) → a
  28. Tangent(xAxis, c) → b, f_1
  29. Line(M, xAxis) → g_1
  30. Intersect(g_1, t) → N
  31. Line(N, p) → h_1
  32. Intersect(h_1, j) → O
  33. Circle(H, O) → c_1
  34. Tangent(j, c_1) → i_1, j_1
  35. Intersect(i_1, j) → P
  36. Intersect(j, i_1) → Q
  37. Intersect(i_1, f) → R
  38. Circle(F, R) → d_1
  39. Intersect(c, g_1) → S, T
  40. Line(S, f) → k_1
  41. Intersect(d_1, k_1) → U, V
  42. Line(T, l) → l_1
  43. Intersect(d_1, l_1) → W, Z
  44. Line(V, j) → m_1
  45. Intersect(m_1, p) → A_1
  46. Circle(H, A_1) → e_1
  47. Tangent(p, e_1) → n_1, p_1
  48. Intersect(g_1, n_1) → B_1
  49. Intersect(g_1, p_1) → C_1
  50. Intersect(m, k) → D_1
  51. Intersect(k, n) → E_1
  52. Intersect(l, m) → F_1
  53. Intersect(n, l) → G_1
  54. Segment(D_1, F_1) → q_1
  55. Segment(F_1, G_1) → r_1
  56. Segment(G_1, E_1) → s_1
  57. Segment(E_1, D_1) → t_1
  58. Intersect(b, q) → H_1
  59. Intersect(b, r) → I_1
  60. Intersect(f_1, q) → J_1
  61. Intersect(f_1, r) → K_1
  62. Segment(H_1, J_1) → a_1
  63. Segment(J_1, K_1) → b_1
  64. Segment(K_1, I_1) → f_2
  65. Segment(I_1, H_1) → g_2
  66. Intersect(g_1, a_1) → L_1
  67. Intersect(g_1, f_2) → M_1
  68. Segment(L_1, M_1) → h_2
  69. Intersect(g_1, h) → N_1
  70. Intersect(g_1, h') → O_1
  71. Segment(N_1, O_1) → i_2
  72. Segment(C, C') → j_2
  73. Segment(J, K) → k_2

AI Drawing Prompt Reference

Draw a geometric construction in one coordinate system to visualize the intersection geometry between a cylinder and a cone. Use two reference axes, and allow a control parameter point to move so the configuration updates. First, determine several key intersection points on the axes and build auxiliary segments to set endpoints. Create circles centered at selected points, then construct tangents between those circles and the relevant lines to obtain constrained tangency points. Use perpendicular helper lines and mirror symmetry to replicate the corresponding parts on the opposite side. Continue by intersecting several auxiliary lines with each other and with the axes to generate additional circle centers and tangency points. Combine the resulting circles, tangents, and intersection points to form the full reconstructed intersection outline.
The prompt above was compiled from the original drawing commands and is for reference only.