MathPlotAI Logo MathPlotAI

Dandelin Spheres: Cone Intersection with Sphere

Dandelin Spheres: Cone Intersection with Sphere
Loading the math board and drawing, please wait…
2026-01-15 03:59
Solid GeometryGeometric ConstructionLocus
In collections Conic Section Collection
This example demonstrates the construction of intersection points between an infinite cone and a plane using Dandelin spheres. Key mathematical concepts include cone geometry, intersection of spheres and cones, and spatial geometry trajectories. Construction involves GeoGebra commands like ConeInfinite, Sphere, IntersectPath, Mirror, and Angle to build cones, spheres, and planes, and perform geometric transformations. Suitable for high school students learning solid geometry or university mathematics courses to understand spatial relationships between cones and spheres and geometric construction techniques.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(zAxis) → B
  3. Line(C, yAxis) → a
  4. Plane(D, a) → b
  5. Mirror(C, D) → C'
  6. Ray(D, C') → c
  7. Angle(c, zAxis) → δ
  8. ConeInfinite(A, B, aa) → d
  9. IntersectPath(b, xz) → int
  10. IntersectPath(xz, d) → g
  11. Intersect(int, g) → V2, V1
  12. Line(A, V1) → c_1
  13. Line(A, V2) → c_2
  14. Line(A, B) → vert
  15. Point(c_2) → K
  16. IntersectPath(b, d) → r
  17. AngularBisector(c_1, int) → f, i
  18. Intersect(vert, f) → O2
  19. Intersect(vert, i) → O1
  20. OrthogonalLine(O2, c_1, xz) → j
  21. OrthogonalLine(O1, c_1, xz) → k
  22. Intersect(c_1, j) → G
  23. Intersect(c_1, k) → H
  24. Circle(O2, G, xz) → p
  25. Circle(O1, H, xz) → q
  26. Intersect(int, q) → F_1, J
  27. Intersect(int, p) → F_2, M
  28. Sphere(O1, F_1) → l
  29. Sphere(O2, F_2) → m
  30. Circle(zAxis, H) → tan2
  31. Circle(zAxis, G) → tan1
  32. Point(r) → Q
  33. Line(Q, A) → gen
  34. Intersect(gen, tan1) → E,
  35. Intersect(gen, tan2) → F,
  36. Segment(Q, F_2) → e
  37. Segment(Q, F) → n
  38. Segment(Q, F_1) → s
  39. Segment(Q, E) → t
  40. Circle(vert, Q) → d_1
  41. Plane(tan2) → o
  42. IntersectPath(o, b) → dir
  43. OrthogonalLine(Q, dir, b) → a_1
  44. Intersect(dir, a_1) → I
  45. OrthogonalLine(Q, a_1, b) → b_1
  46. Intersect(b_1, int) → L
  47. Intersect(dir, r) → N,
  48. Intersect(int, r) → O, P
  49. Intersect(dir, int) → R
  50. Segment(Q, I) → e_1
  51. Segment(L, R) → f_1
  52. Intersect(b_1, r) → S, T
  53. PointIn(b) → W
  54. OrthogonalPlane(R, dir) → u
  55. IntersectPath(u, d) → g_1
  56. Intersect(g_1, tan2) → U, V, ,
  57. Intersect(g_1, d_1) → Z, A_1, ,
  58. Segment(Z, U) → h_1
  59. If(bb < aa, text3, If[bb > aa, text1, text2]) → angle
  60. Angle(Q, I, R, b) → α
  61. Point(dir) → B_1
  62. Angle(F_1, R, B_1, b) → β

AI Drawing Prompt Reference

Draw a 3D solid geometry diagram: an infinite cone with apex at the origin and vertical axis. A slanted plane intersects the cone, producing an elliptical cross-section. Inside the cone, place two Dandelin spheres, each tangent to the cone's lateral surface and to the cutting plane; their tangency points with the plane are the ellipse's foci. From any point on the ellipse, draw segments to both foci and to the cone's apex, illustrating that the sum of distances to the foci is constant. Also show the circles where the spheres touch the cone, plus auxiliary lines connecting sphere centers and tangency points, to clarify the spatial relationships among cone, plane, and spheres.
The prompt above was compiled from the original drawing commands and is for reference only.