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3D Conic Sections: Dandelin Spheres

3D Conic Sections: Dandelin Spheres
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2020-02-17 13:46
CircleCoordinate GeometrySolid GeometryGeometric ConstructionLocus
① This GeoGebra example demonstrates conic sections (ellipse, parabola, hyperbola) as intersections of a plane with an infinite cone, using Dandelin spheres to reveal their focal properties. ② Core mathematical concepts include the geometric definition of conics, foci and directrices, tangent spheres and planes, and spatial relationships such as intersection, symmetry, and angle bisectors. ③ Construction highlights: the InfiniteCone command creates the cone, IntersectPath computes the section curve, Sphere constructs the Dandelin spheres, and a rich set of commands (Point, Line, Plane, Mirror, Ray, Angle, AngularBisector, OrthogonalLine, Circle, Segment, PointIn, OrthogonalPlane, If) build auxiliary elements like vertices V1, V2, sphere centers O1, O2, foci F_1, F_2, and various helper points, lines, and planes. ④ Suitable for high school or early college courses in solid geometry and analytic geometry, as well as for mathematical demonstrations illustrating the unified geometric nature 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) → δ
  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
  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