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Ellipse Animation with Foci

Ellipse Animation with Foci
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2016-06-14 23:09
VectorCoordinate GeometryGeometric Construction
This example demonstrates the relationship between an ellipse's foci and vertices through animation. It involves core concepts like ellipse definition, foci, vertices, Pythagorean theorem, and vector operations. Construction highlights include commands like Focus, Vertex, Segment, Translate, Vector, and Intersect. Suitable for high school mathematics education to understand elliptical properties and algebraic expressions.

GeoGebra Commands & Parameters

  1. Focus(c) → F_1, F_2
  2. Vertex(c) → C, D, E, F
  3. Segment(C, D) → d
  4. Translate(C, (0, b + 4)) → C'
  5. Translate(D, (0, b + 4)) → D'
  6. Point(c) → G
  7. Segment(G, F_1) → e
  8. Segment(G, F_2) → f
  9. Translate(C, (b + 4, 0)) → C'_1
  10. Vector(C, C') → u
  11. Vector(D, D') → v
  12. Dilate(C, If[0 < n < 1, 1 - n, 0], C') → C'_2
  13. Dilate(D, If[0 < n < 1, 1 - n, 0], D') → D'_2
  14. Dilate(F_1, If[1 < n < 2, 2 - n, 1], C') → A'
  15. Vector(F_1, A') → w
  16. Translate(G, w) → G'
  17. Segment(A', G') → m
  18. Angle(F_2, F_1, G) → α
  19. Rotate(A', If[1 < n < 2, ((-(2 - n)) * α)], A') → A''
  20. Rotate(A', If[1 < n < 2, ((-(n - 2)) * α), 0], A'') → A''_1
  21. Rotate(A', If[1 < n < 2, ((-(n - 1)) * α), (-α)], A') → A''_2
  22. Rotate(G', If[1 < n < 2, ((-(n - 1)) * α), (-α)], A') → G''
  23. Circle(C', e) → q
  24. Segment(F_1, G) → s
  25. Dilate(F_2, If[2 < n < 3, 3 - n, 1], D') → B'
  26. Vector(F_2, B') → t
  27. Translate(G, t) → G'_2
  28. Translate(F_2, t) → B'_1
  29. Segment(G'_2, B'_1) → f'
  30. Angle(G, F_2, F_1) → β
  31. Rotate(G'_2, If[2 < n < 3, ((n - 2) * β)], B') → L
  32. Rotate(B'_1, If[2 < n < 3, ((n - 2) * β)], B') → M
  33. Segment(G, F_2) → b_1
  34. Center(c) → A
  35. Segment(A, G) → c_1
  36. Segment(A, G) → d_1
  37. Dilate(A, If[1 < n < 2, 2 - n, 0], C') → A'_1
  38. Vector(F_1, A'_1) → u_1
  39. Translate(A, u_1) → A'_2
  40. Angle(D, F_1, G) → γ
  41. Rotate(A'_2, ((-(n - 1)) * γ), A'_2) → B
  42. Circle(C, e) → p
  43. Intersect(p, d, 1) → H
  44. Dilate(F_1, If[0 < n < 1, 1 - n, 1], C) → I
  45. Vector(F_1, I) → l
  46. Translate(G, l) → G'_1
  47. Translate(F_1, l) → J
  48. Segment(G'_1, J) → e'
  49. Rotate(G'_1, If[0 < n < 1, (-((n * α))), (-α)], I) → K
  50. Rotate(J, If[0 < n < 1, (-((n * α))), (-α)], I) → J'
  51. Segment(K, J') → e''
  52. Segment(C, H) → r
  53. Dilate(F_2, If[1 < n < 2, 2 - n, 1], D) → N
  54. Vector(F_2, N) → v_1
  55. Translate(G, v_1) → G'_3
  56. Translate(F_2, v_1) → O
  57. Segment(G'_3, O) → f'_1
  58. Rotate(G'_3, If[1 < n < 2, ((n - 1) * β), β], N) → P
  59. Rotate(O, If[1 < n < 2, ((n - 1) * β), β], N) → O'
  60. Segment(P, O') → a_1
  61. Segment(D, H) → e_1
  62. Dilate(F_1, 1 - n, C) → Q
  63. Vector(F_1, Q) → h
  64. Translate(A, h) → A'_3
  65. Translate(G, h) → G'_4
  66. Segment(A'_3, G'_4) → i
  67. Angle(D, Center[c], G) → δ
  68. Rotate(A'_3, If[0 < n < 1, (-((n * δ)))], Q) → R
  69. Rotate(G'_4, If[0 < n < 1, (-((n * δ)))], Q) → S
  70. Segment(R, S) → i'
  71. Dilate(F_2, If[1 < n < 2, 2 - n], D) → T
  72. Vector(F_1, T) → w_1
  73. Translate(A, w_1) → A'_4
  74. Translate(G, w_1) → G'_5
  75. Segment(A'_4, G'_5) → f_1
  76. Angle(G, Center[c], C) → ε
  77. Rotate(A'_4, If[1 < n < 2, ((n - 1) * ε)], A'_4) → U
  78. Rotate(G'_5, If[1 < n < 2, ((n - 1) * ε)], A'_4) → V
  79. Segment(U, V) → g_1