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Dandelin Spheres - Construction and Eccentricity

Dandelin Spheres - Construction and Eccentricity
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2016-04-08 11:36
Coordinate GeometrySolid GeometryGeometric ConstructionSlider AnimationLocus
This GeoGebra example demonstrates the classical Dandelin spheres construction in 3D space, visually linking the conic sections (ellipse, hyperbola, parabola) to their geometric definition and eccentricity. Core mathematical concepts include the intersection of a plane with an infinite cone, tangency of spheres to the cone and plane, foci and directrix properties, and eccentricity as a measure of curve shape. The construction uses commands such as InfiniteCone, IntersectPath, Sphere, Point, Line, Plane, Mirror, Ray, Angle, AngularBisector, OrthogonalLine, Circle, Segment, and PointIn to build the two tangent spheres and the cutting plane. By adjusting the plane's angle, learners can dynamically observe how eccentricity changes and how the conic type transitions. Suitable for high school or undergraduate courses in analytic geometry, solid geometry, and for anyone interested in the unified theory of conic sections.

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) → β
  8. InfiniteCone(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_1
  24. Circle(O2, G, xz) → p
  25. Circle(O1, H_1, 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_1) → 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. Plane(tan2) → o
  41. IntersectPath(o, b) → dir
  42. OrthogonalLine(Q, dir, b) → a_1
  43. Intersect(dir, a_1) → I
  44. OrthogonalLine(Q, a_1, b) → b_1
  45. Intersect(dir, r) → N,
  46. Intersect(int, r) → O, P
  47. Intersect(dir, int) → R
  48. Intersect(b_1, r) → S, T
  49. PointIn(b) → W
  50. If(bb < aa, text3, If[bb > aa, text1, text2]) → angle
  51. Angle(Q, I, R, b) → α
  52. Point(dir) → B_1
  53. OrthogonalLine(Q, dir, space) → i_1
  54. Intersect(dir, i_1) → C_1
  55. Segment(Q, C_1) → dirDist
  56. Intersect(i_1, dirDist) → D_1
  57. OrthogonalLine(Q, o) → j_1
  58. Intersect(j_1, o) → H
  59. Segment(Q, H) → k_1
  60. Angle(F, H, Q) → ε
  61. Segment(F, H) → l_1
  62. Segment(H, C_1) → m_1
  63. Angle(H, Q, F) → δ
  64. Angle(C_1, H, Q) → ζ
  65. Angle(H, Q, C_1) → η
  66. Angle(B, A, Q) → θ
  67. Angle(R, I, Q) → β_1