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Dandelin Sphere Demonstration

Dandelin Sphere Demonstration
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2018-05-07 11:06
Solid GeometryTransformationGeometric Construction
In collections GeoGebra 3D & AR: PreCalc & Calculus Resources
This example demonstrates the dynamic process of Dandelin spheres. Key mathematical concepts include the intersection of cones and spheres in solid geometry. Using various GeoGebra commands such as constructing cones, spheres, and rotations, it visually illustrates how Dandelin spheres help understand the properties of conic sections. Suitable for high school geometry and solid geometry courses.

GeoGebra Commands & Parameters

  1. Cone(B_2, A_2, r) → a, c, b
  2. Segment((0, 0, 0), (0, 0, 10)) → f
  3. Point(f) → A
  4. Segment(A, (0, 0, 10)) → g
  5. Line(A_2, C) → i
  6. Distance(A, i) → d
  7. Sphere(A, d) → e
  8. PointIn(e) → H
  9. Line(A, H) → j
  10. Vector(A, H) → u
  11. OrthogonalLine(H, u, xOyPlane) → k
  12. Rotate(k, π / 2, j) → k'
  13. Sphere(H, 2) → l
  14. Intersect(k', l) → I, J
  15. Intersect(k, l) → K, L
  16. Plane(J, L, K) → m
  17. IntersectPath(m, a) → p
  18. Focus(p) → O, P
  19. OrthogonalLine(P, m) → q
  20. Line((0, 0, 0), (0, 0, 1)) → s
  21. Intersect(s, q) → B
  22. Sphere(B, P) → o
  23. Point(p) → M
  24. Vertex(p) → N, R, S, T
  25. IntersectPath(t, o) → c_1
  26. IntersectPath(v, e) → d_1
  27. Line(A_2, M) → f_1
  28. Intersect(f_1, e) → U, V
  29. Intersect(f_1, o) → W, Z
  30. Dilate(A_2, If[0 < w ≤ 1, 1 - w, 0], U) → A'
  31. Segment(A_2, A') → g_1
  32. Dilate(U, If[1 < w ≤ 2, 2 - w, 0], M) → U'
  33. Segment(U, U') → h_1
  34. Dilate(M, If[2 < w ≤ 3, 3 - w, 0], W) → M'
  35. Segment(M, M') → i_1
  36. Angle(H, M, U) → α
  37. Rotate(U, If[4 < w ≤ 5, ((w - 4) * α), α], M, Plane[M, U, H]) → U'_1
  38. Rotate(U', If[4 < w ≤ 5, ((w - 4) * α), α], M, Plane[M, U, H]) → U''
  39. Segment(U'_1, U'') → j_1
  40. Angle(P, M, W) → β
  41. Rotate(M, If[5 < w ≤ 6, ((-(w - 5)) * β), (-β)], M, Plane[P, W, M]) → M'_1
  42. Rotate(M', If[5 < w ≤ 6, ((-(w - 5)) * β), (-β)], M, Plane[P, W, M]) → M''
  43. Segment(M'_1, M'') → k_1
  44. Line(N, R) → l_1
  45. Sphere(N, j_1) → a_1
  46. Intersect(l_1, a_1) → A_1, B_1
  47. Dilate(H, If[7 < w ≤ 8, 8 - w, 0], N) → H'
  48. Vector(H, H') → u_1
  49. Translate(U'_1, u_1) → C_1
  50. Translate(U'', u_1) → U'''
  51. Segment(C_1, U''') → m_1
  52. Angle(M, H, R) → γ
  53. Rotate(C_1, If[7 < w ≤ 8, ((-(w - 7)) * γ), (-γ)], H', Plane[H', R, M]) → D_1
  54. Rotate(U''', If[7 < w ≤ 8, ((-(w - 7)) * γ), (-γ)], H', Plane[H', R, M]) → E_1
  55. Segment(D_1, E_1) → n_1
  56. Dilate(P, If[8 < w ≤ 9, 9 - w, 0], R) → P'
  57. Vector(P, P') → v_1
  58. Translate(M'_1, v_1) → F_1
  59. Translate(M'', v_1) → M'''
  60. Segment(F_1, M''') → p_1
  61. Angle(N, P, M) → δ
  62. Rotate(F_1, If[8 < w ≤ 9, ((w - 8) * δ), δ], P', Plane[N, R, M]) → G_1
  63. Rotate(M''', If[8 < w ≤ 9, ((w - 8) * δ), δ], P', Plane[N, R, M]) → H_1
  64. Segment(G_1, H_1) → q_1

AI Drawing Prompt Reference

Create a 3D geometric demonstration showing a cone intersected by a plane, with two spheres (Dandelin spheres) tangent to the cone and the plane. The cone has a vertical axis, and the plane cuts through it forming an ellipse. The two spheres are placed inside the cone, each tangent to the cone's lateral surface and to the plane, with their tangency points on the plane being the foci of the ellipse. Include auxiliary lines, segments, and dynamic elements such as rotating points and moving segments to illustrate the construction. Show the cone, spheres, plane, ellipse, and relevant points, lines, and angles in a clear spatial arrangement.
The prompt above was compiled from the original drawing commands and is for reference only.