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Dandelin Spheres and Parabola

Dandelin Spheres and Parabola
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2020-02-17 11:18
Coordinate GeometrySolid GeometryGeometric Construction
This example demonstrates how Dandelin spheres explain why geometrically defined parabola, ellipse, and hyperbola are the same curves as conic sections. It involves core concepts like conic sections, Dandelin spheres, geometric definitions, and cone sections. Construction uses GeoGebra commands such as InfiniteCone, Rotate, IntersectPath, Circle, and Sphere. Suitable for high school students and educators seeking a deeper geometric understanding of conic sections.

GeoGebra Commands & Parameters

  1. InfiniteCone((0, 0, 0), zAxis, α) → a
  2. Axes(a) → d, e, f
  3. Rotate(f, (-α), yAxis) → lineOA
  4. Point(lineOA) → A
  5. Rotate(f, α, yAxis) → f'
  6. Point(f') → B
  7. Plane(A, B_1, B) → b
  8. IntersectPath(b, a) → EllipseHyperbola
  9. Axes(EllipseHyperbola) → i, j
  10. AngularBisector(f', i) → b2, b1
  11. Center(a) → O
  12. Vector(O, B) → vOB
  13. Angle(Vector[A, B], Vector[A, O]) → β
  14. Vector(O, A) → vOA
  15. Rotate(vOA, β / 2, A, k) → vOA'
  16. Angle(Vector[B, A], Vector[O, B]) → α1
  17. Rotate(vOB, α1 / 2, B, k) → vOB'
  18. Ray(B, B') → g
  19. Intersect(f, b1) → CenterSphere1
  20. OrthogonalLine(CenterSphere1, i, k) → h
  21. Intersect(i, h) → H
  22. Circle(CenterSphere1, H, k) → p
  23. Sphere(CenterSphere1, H) → l
  24. Ray(A, A') → m
  25. Intersect(f, b2) → CenterSphere2
  26. OrthogonalLine(CenterSphere2, i, k) → n
  27. Intersect(i, n) → M
  28. Circle(CenterSphere2, M, k) → s
  29. Point(EllipseHyperbola) → P
  30. Line(O, P) → a_1
  31. Segment(M, P) → b_1
  32. Segment(P, H) → e_1
  33. Sphere(CenterSphere2, M) → l'
  34. OrthogonalLine(CenterSphere2, lineOA, k) → q
  35. OrthogonalLine(CenterSphere1, lineOA, k) → r
  36. Intersect(lineOA, r) → C
  37. Intersect(q, lineOA) → D
  38. Circle(f, C) → tangentCircle1
  39. Circle(f, D) → tangentCircle2
  40. Line(O, P) → lineOP
  41. Intersect(lineOP, tangentCircle2) → TangencyP2,
  42. Intersect(lineOP, tangentCircle1) → TangencyP1,
  43. Segment(P, TangencyP1) → PtoTangent1
  44. Segment(TangencyP2, P) → PtoTangent2
  45. Segment(O, TangencyP2) → coneEdge
  46. Tangent(A, EllipseHyperbola) → t,
  47. Plane(t, lineOA) → tangentToConePlane
  48. Plane(B, tangentToConePlane) → planeForParabola
  49. IntersectPath(planeForParabola, a) → Parabola
  50. Intersect(f, Parabola) → E,
  51. Intersect(f, planeForParabola) → F_1
  52. Line(B, F_1) → c
  53. AngularBisector(f', c) → f_1, g_1
  54. Intersect(g_1, f) → C_{SphereForParabola}
  55. OrthogonalLine(C_{SphereForParabola}, c, space) → h_1
  56. Intersect(h_1, c) → F
  57. Sphere(C_{SphereForParabola}, F) → o
  58. Point(Parabola) → G
  59. Segment(F, G) → i_1
  60. Intersect(f', o) → I, J
  61. Circle(f, I) → c_1
  62. Sphere(B, F) → u
  63. Intersect(c, u) → K, L
  64. OrthogonalPlane(K, c) → p_1
  65. IntersectPath(p_1, planeForParabola) → directrix
  66. OrthogonalLine(G, p_1) → k_1
  67. Intersect(directrix, Parabola) → N, T
  68. Intersect(directrix, k_1) → Q
  69. Segment(G, O) → l_1
  70. Intersect(l_1, c_1, 1) → R
  71. Plane(c_1) → tangentCirclePlaneForParabola
  72. OrthogonalLine(G, tangentCirclePlaneForParabola) → m_1
  73. Intersect(m_1, tangentCirclePlaneForParabola) → S
  74. Angle(S, G, R) → γ
  75. Angle(Q, G, S) → δ
  76. Segment(G, R) → j_1
  77. Segment(Q, G) → n_1
  78. Segment(Q, S) → q_1
  79. Segment(S, R) → r_1
  80. Angle(G, S, Q) → ε
  81. Angle(R, S, G) → ζ
  82. If(checkBoxEllHyp, checkBoxParabola + (0 * z) = false) → p_2
  83. If(checkBoxEllHyp, checkBoxParabola + (0 * z) = false) → q_2
  84. If(checkBoxEllHyp, checkBoxParabola + (0 * z) = false) → r_2
  85. If(checkBoxEllHyp, checkBoxParabola + (0 * z) = false) → p_3
  86. If(checkBoxEllHyp, checkBoxParabola + (0 * z) = false) → q_3
  87. If(checkBoxEllHyp, checkBoxParabola + (0 * z) = false) → r_3
  88. If(checkBoxEllHyp, checkBoxParabola + (0 * z) = false) → p_4
  89. If(checkBoxEllHyp, checkBoxParabola + (0 * z) = false) → q_4
  90. If(checkBoxEllHyp, checkBoxParabola + (0 * z) = false) → r_4
  91. If(checkBoxEllHyp, checkBoxParabola + (0 * z) = false) → p_5
  92. If(checkBoxEllHyp, checkBoxParabola + (0 * z) = false) → q_5
  93. Angle(G, Q, K) → η

AI Drawing Prompt Reference

Draw a cone with a rotating line to create a conic section. Show two Dandelin spheres tangent to the curve at two and one point respectively. Label the cone's vertex, points on the curve, tangent points, and centers of the Dandelin spheres. Draw the spherical surfaces tangent to the curve. Finally, connect the vertex to a point on the curve to form a line.
The prompt above was compiled from the original drawing commands and is for reference only.