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Triangle Rotation to Conic Sections Animation

Triangle Rotation to Conic Sections Animation
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2017-01-12 03:29
TriangleCircleCoordinate GeometryGeometric Construction
In collections CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2
This example demonstrates the generation of conic sections through the rotation of an equilateral triangle, visually linking geometric transformations with analytical geometry. Core concepts include equilateral triangle construction, rotation transformations, formation of conic sections (ellipse, parabola, hyperbola), linear functions, and confidence intervals. Key GeoGebra commands and objects used are Rotate, Polygon, Circle, Intersect, and Dilate. Suitable for high school and above learners to understand geometric transformations in analytical geometry and the principles of conic section generation.

GeoGebra Commands & Parameters

  1. Rotate(A, 60°, B) → A'
  2. Angle(A, B, A') → α
  3. Ray(B, A) → f
  4. Ray(B, A') → g
  5. Point(f) → C
  6. Point(g) → D
  7. Polygon(C, D, B) → poly1, b, c, d
  8. Rotate(D, 60°, B) → D'
  9. Angle(D, B, D') → β
  10. Rotate(C, (-(60°)), B) → C'
  11. Angle(C', B, C) → γ
  12. Line(C', D') → h
  13. Segment(B, C) → i
  14. Segment(B, D) → j
  15. Min(i, j) → a
  16. Segment(C, D) → k
  17. Min(i, k) → l
  18. Min(a, l) → m
  19. Circle(B, (0.2 * m)) → e
  20. Intersect(e, h, 1) → E
  21. Intersect(e, i, 1) → F
  22. Intersect(e, j, 1) → G
  23. Intersect(e, h, 2) → H
  24. Angle(D, C, B) → δ
  25. Angle(B, D, C) → ε
  26. Rotate(D, δ / 2, C) → D'_1
  27. Angle(D, C, D'_1) → ζ
  28. Ray(C, D'_1) → p
  29. Rotate(B, ε / 2, D) → B'
  30. Angle(B, D, B') → η
  31. Ray(D, B') → q
  32. Circle(C, (0.2 * m)) → r
  33. Circle(D, (0.2 * m)) → s
  34. Intersect(r, p, 1) → I
  35. Intersect(r, k, 1) → J
  36. Intersect(s, j, 1) → K
  37. Intersect(s, q, 1) → L
  38. Intersect(i, q) → M
  39. Intersect(j, p) → N
  40. CircleSector(B, E, F) → t
  41. Rotate(t, If[0 < n ≤ 1, (n * π / 3), π / 3], B) → t'
  42. Rotate(t', If[2 < n ≤ 3, ((n - 2) * π / 3), π / 3], B) → t''
  43. Rotate(t'', If[4 < n ≤ 5, ((-(n - 4)) * π / 3), ((-π)) / 3], B) → t'''
  44. Dilate(C, If[5 < n ≤ 6, 6 - n, 0], N) → C'_1
  45. Segment(C, C'_1) → f_1
  46. Dilate(D, If[6 < n ≤ 7, 7 - n, 0], M) → D'_2
  47. Segment(D, D'_2) → g_1
  48. Intersect(r, i, 1) → O
  49. Intersect(s, k, 1) → P
  50. CircleSector(C, I, O) → c_1
  51. CircleSector(D, L, P) → d_1
  52. Rotate(c_1, If[7 < n ≤ 8, ((-(n - 7)) * ζ), (-ζ)], C) → e_1
  53. Rotate(d_1, If[8 < n ≤ 9, ((-(n - 8)) * η), (-η)], D) → h_1
  54. Dilate(C, If[9 < n ≤ 10, 10 - n, 0], M) → C'_2
  55. Segment(C, C'_2) → i_1
  56. Dilate(D, If[10 < n ≤ 11, 11 - n, 0], N) → D'_3
  57. Segment(D, D'_3) → j_1
  58. Rotate(C, If[11 < n ≤ 12, ((-(n - 11)) * δ), (-δ)], C) → C'_3
  59. Rotate(C'_2, If[11 < n ≤ 12, ((-(n - 11)) * δ), (-δ)], C) → Q
  60. Segment(C'_3, Q) → k_1
  61. Rotate(M, If[11 < n ≤ 12, ((-(n - 11)) * δ), (-δ)], C) → M'
  62. Rotate(D, If[12 < n ≤ 13, ((n - 12) * ε), ε], D) → D'_4
  63. Rotate(D'_3, If[12 < n ≤ 13, ((n - 12) * ε), ε], D) → R
  64. Segment(D'_4, R) → l_1

AI Drawing Prompt Reference

Draw an equilateral triangle and rotate it around one vertex to generate ellipses, parabolas, and hyperbolas. First, construct an equilateral triangle, then rotate one of its vertices around a fixed point to form a new triangle. Next, use rotation and intersection operations to gradually generate ellipses, parabolas, and hyperbolas. Finally, adjust parameters to dynamically show the changes in these conic sections.
The prompt above was compiled from the original drawing commands and is for reference only.