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Locus of Double Branch Hyperbola Demo

Locus of Double Branch Hyperbola Demo
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2019-02-11 01:36
FunctionTrigonometryCoordinate GeometryLocus
This example demonstrates the locus generation of a double branch hyperbola. It involves core mathematical concepts such as hyperbola definition, foci, vertices, vector transformations, etc. The construction utilizes GeoGebra commands and objects like Focus, Vertex, Segment, Translate, Point, Vector, Dilate, Angle, Rotate, Circle, Intersect, Center as outlined in the summary. Suitable for high school and above students to understand the geometric properties and transformation applications of hyperbolas.

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

AI Drawing Prompt Reference

Draw a double hyperbola locus. First, define the foci F1 and F2, and the vertices C, D, E, F. Then, use transformations like translation, rotation, and scaling to construct the hyperbola's locus. This involves translating points C and D, creating vectors u and v, performing scaling transformations on points C and D, constructing segments, angles, circles, and intersections, and ultimately forming the hyperbola's locus.
The prompt above was compiled from the original drawing commands and is for reference only.