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Animation 30: Hyperbola, Matrix, and Cuboid Transformations

Animation 30: Hyperbola, Matrix, and Cuboid Transformations
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2016-06-16 10:56
MatrixTransformationGeometric ConstructionSlider Animation
① This GeoGebra example, titled 'Animation 30', is a dynamic demonstration that integrates hyperbolas, matrix transformations, cuboids, quadrilaterals, secant lines, and set-theoretic concepts in a single geometric construction. ② Core mathematical concepts include foci and vertices of a hyperbola, linear transformations (translation, scaling, rotation) applied via matrices, properties of secant lines, quadrilaterals and cuboids in space, and the set-theoretic organization of geometric objects. ③ Construction highlights: from the summary, commands such as Focus, Vertex, Segment, Translate, Point, Vector, Dilate, Angle, Rotate, Circle, Intersect, and Center are used to create foci F_1, F_2, vertices C, D, E, F, and derived points (C', D', C'_1, C'_2, A'', etc.) through translations, rotations, and dilations, along with auxiliary lines (secants d, e, f, g, h, i, j, k, n, u, v, l, m, p, q) and slider-driven animation. ④ Suitable for high school to undergraduate mathematics education, this example helps visualize conic sections and linear transformations, and serves as a reference for designing dynamic geometry lessons.

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. Segment(C'_2, D'_2) → l
  15. Dilate(F_1, If[1 < n < 2, 2 - n, 1], C') → A'
  16. Vector(F_1, A') → w
  17. Translate(G, w) → G'
  18. Segment(A', G') → m
  19. Segment(C, D) → p
  20. Angle(F_2, F_1, G) → α
  21. Rotate(A', If[1 < n < 2, ((-(2 - n)) * α)], A') → A''
  22. Rotate(A', If[1 < n < 2, ((-(n - 2)) * α), 0], A'') → A''_1
  23. Rotate(A', If[1 < n < 2, ((-(n - 1)) * α), (-α)], A') → A''_2
  24. Rotate(G', If[1 < n < 2, ((-(n - 1)) * α), (-α)], A') → G''
  25. Segment(A''_2, G'') → m'
  26. Circle(C', e) → q
  27. Intersect(q, l, 1) → H
  28. Segment(C', H) → r
  29. Segment(F_1, G) → s
  30. Dilate(F_2, If[2 < n < 3, 3 - n, 1], D') → B'
  31. Vector(F_2, B') → t
  32. Translate(G, t) → G'_1
  33. Translate(G, t) → G'_2
  34. Translate(F_2, t) → B'_1
  35. Segment(G'_2, B'_1) → f'
  36. Angle(G, F_2, F_1) → β
  37. Segment(H, D') → a_1
  38. Segment(G, F_2) → b_1
  39. Center(c) → A
  40. Dilate(A, If[1 < n < 2, 2 - n, 0], C') → A'_1
  41. Vector(F_1, A'_1) → u_1
  42. Translate(A, u_1) → A'_2
  43. Translate(G, u_1) → G'_3
  44. Segment(A'_2, G'_3) → e_1
  45. Angle(D, F_1, G) → γ
  46. Rotate(A'_2, ((-(n - 1)) * γ), A'_2) → B
  47. Rotate(G'_3, ((-(n - 1)) * γ), A'_2) → I
  48. Dilate(G, If[2 < n < 3, 3 - n, 0], D') → G'_4
  49. Vector(G, G'_4) → v_1
  50. Translate(A, v_1) → A'_3
  51. Translate(G, v_1) → G'_5
  52. Segment(A'_3, G'_5) → g_1
  53. Rotate(A'_3, If[2 < n < 3, ((-(n - 2)) * γ)], G'_4) → J
  54. Rotate(G'_5, If[2 < n < 3, ((-(n - 2)) * γ)], G'_4) → K
  55. Circle(C', Distance[A''_2, G'']) → k_1
  56. Intersect(k_1, i_1, 2) → N
  57. Segment(C', N) → j_1
  58. Dilate(F_2, If[2 < n < 3, 3 - n], N) → O
  59. Vector(F_2, O) → w_1
  60. Translate(G, w_1) → G'_6
  61. Translate(F_2, w_1) → L
  62. Segment(G'_6, L) → f'_1
  63. Circle(N, f) → p_1
  64. Rotate(G'_6, If[2 < n < 3, ((n - 2) * β)], O) → M
  65. Rotate(L, If[2 < n < 3, ((n - 2) * β)], O) → L'
  66. Segment(M, L') → l_1
  67. Segment(N, D') → m_1
  68. Dilate(C', If[3 < n ≤ 4, 4 - n], C) → C''
  69. Dilate(D', If[3 < n ≤ 4, 4 - n], D) → D''
  70. Segment(C'', D'') → n_1