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Conics Generated Inside a Cone

Conics Generated Inside a Cone
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2021-01-18 16:09
CircleSolid GeometryTransformationGeometric ConstructionLocus
This example demonstrates how different cutting planes intersect a single cone to produce conic sections (e.g., ellipse/parabola/hyperbola) and visualizes the geometric relationships and symmetries behind the construction. The key mathematical ideas include the cone-and-plane definition of conics, intersection geometry between a cone surface and a plane, and the use of symmetry via mirror transformations to link corresponding elements. Construction highlights in GeoGebra: use Cone to build the 3D solid and Plane to define the cutting plane; apply Mirror to create symmetric counterparts; employ IntersectPath (and the associated intersection objects) to obtain the curve lying on the cone from the plane cut; use auxiliary primitives such as Circle, Line, Point, Segment, and Polygon to set up reference geometry and boundary conditions. The listed objects (e.g., key points/parameters and sets used to assemble the curve like m_1, A, F/F', R, and auxiliary constructions such as G/H/I, poly1) organize the curve from computed intersection results. Suitable for secondary to early university teaching and for hands-on geometric construction practice of conic sections.

GeoGebra Commands & Parameters

  1. Mirror(F, A) → F'
  2. Circle(F, R, zAxis) → c
  3. Cone(c, (-h)) → a, d, b
  4. Mirror(a, A) → a'
  5. Segment(F, F') → f
  6. Polygon(H, I, 4, xOyPlane) → poly1, g, i, j, k, J, K
  7. Plane(I, H, G) → p
  8. IntersectPath(p, a) → e
  9. IntersectPath(p, a') → q
  10. Mirror(A, zAxis) → A'
  11. Line(A, A') → l
  12. Point(l) → B
  13. Point(l) → C
  14. Point(l) → D
  15. Point(l) → E
  16. Mirror(L, B) → L'
  17. Mirror(M, C) → M'
  18. Mirror(N, D) → N'
  19. Mirror(O, E) → O'
  20. Circle(L, R, zAxis) → r
  21. Circle(M, R, zAxis) → s
  22. Circle(N, R, zAxis) → t
  23. Circle(O, R, zAxis) → c_1
  24. Cone(r, (-h)) → m, d_1, n
  25. Cone(s, (-h)) → o, e_1, u
  26. Cone(t, (-h)) → v, f_1, w
  27. Cone(c_1, (-h)) → a_1, g_1, b_1
  28. Mirror(L, B) → L'_{1}
  29. Mirror(M, C) → M'_{1}
  30. Segment(L, L'_{1}) → i_1
  31. Segment(M, M'_{1}) → j_1
  32. Segment(N, N') → k_1
  33. Segment(O, O') → l_1
  34. Mirror(m, B) → m'
  35. Mirror(o, C) → o'
  36. Mirror(v, D) → v'
  37. Mirror(a_1, E) → h_1
  38. Plane(Q, S, P) → p_1
  39. Mirror(S, P) → S'
  40. Mirror(Q, P) → Q'
  41. IntersectPath(p_1, m) → q_1
  42. IntersectPath(p_1, m') → r_1
  43. Polygon(Q', S', Q, S) → q1, q', s', q_2, s_1
  44. Mirror(U, T) → U'
  45. Mirror(V, T) → V'
  46. Plane(U, V, V') → p_2
  47. IntersectPath(p_2, o') → t_1
  48. IntersectPath(p_2, o) → c_2
  49. Polygon(V, U, V', U') → q2, v_1, u_1, n_1, u'
  50. Mirror(Z, W) → Z'
  51. Mirror(A_1, W) → B_1
  52. Plane(Z, A_1, W) → r_2
  53. IntersectPath(r_2, v') → d_2
  54. IntersectPath(r_2, v) → e_2
  55. Polygon(A_1, Z, B_1, Z') → q3, f_2, z_1, g_2, z'
  56. Mirror(D_1, l_1) → F_1
  57. Plane(D_1, F_1, C_1) → p_3
  58. IntersectPath(p_3, h_1) → h_2
  59. IntersectPath(p_3, b_1) → k_2
  60. Mirror(D_1, C_1) → G_1
  61. Mirror(F_1, C_1) → H_1
  62. Polygon(F_1, D_1, H_1, G_1) → q4, i_2, j_2, l_2, m_2

AI Drawing Prompt Reference

Create a 3D scene with a circular base and use it to generate a cone. Establish symmetry by mirroring key circular and cone elements across a vertical axis, producing corresponding reflected counterparts. Then define an oblique cutting plane using three points and slice the cone with this plane, obtaining the conic intersection curve(s) on the cone surface. Repeat the intersection for the mirrored cone setup and mirror-related plane configuration so the resulting curves and supporting reference segments/lines appear pairwise symmetric. Add appropriate circles, lines, and points to mark the geometric relations and the symmetry between the two conic sections.
The prompt above was compiled from the original drawing commands and is for reference only.