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Billiards: Vector and Geometric Transformations

Billiards: Vector and Geometric Transformations
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2014-10-05 06:17
VectorTrigonometryCoordinate GeometryGeometric Construction
This example demonstrates vector and geometric transformations in billiards. By setting the positions, speeds, and angles of the balls, one can observe the trajectories and collision effects of the balls. The core mathematical concepts involved include vectors, angles, and geometric transformations. The construction uses commands and objects such as Vector, Polygon, PointIn, Segment, Angle, If, Translate, Circle, Line, OrthogonalLine, Intersect, Distance, Mirror in GeoGebra. It is suitable for high school students learning vectors, geometric transformations, and kinematics in physics.

GeoGebra Commands & Parameters

  1. Vector(A, B) → u
  2. Polygon(Θ, Κ, Λ, Μ) → πολύγ1, θ, κ, λ, μ
  3. PointIn(πολύγ1) → E
  4. Segment(C, E) → f
  5. Angle(E, C, B) → β
  6. If(0 ≤ β < 2° ∨ 360° ≥ β > 358°, (f - 0.8) / abs(u), If[β ≥ 2° ∧ β ≤ 3° ∨ β ≥ 357° ∧ β ≤ 358°, (f - 0.75) / abs(u), (f - 0.7) / abs(u)]) → t1
  7. Translate(D, u) → D'
  8. Vector(D, D') → v
  9. Segment(DD, E) → h
  10. Segment(D, C) → g
  11. Circle(D, 0.4) → k
  12. Circle(E, 0.4) → p
  13. If(h ≤ 0.8, j ≟ true, j ≟ false) → jj
  14. Angle(D, E, C) → α
  15. Line(C, D) → m
  16. Line(DD, E) → a
  17. OrthogonalLine(D', a) → b
  18. OrthogonalLine(D, a) → c_1
  19. Intersect(a, c_1) → Α
  20. Intersect(a, b) → Β
  21. Vector(Α, Β) → wt1b
  22. OrthogonalLine(A, a) → d
  23. OrthogonalLine(B, a) → e
  24. Intersect(a, e) → F
  25. Intersect(a, d) → G
  26. Vector(F, G) → wt1r
  27. Line(C, D') → i
  28. OrthogonalLine(E, i) → l
  29. Intersect(i, l) → H
  30. Segment(E, H) → n
  31. Segment(EE, TT) → r
  32. Line(DD, EE) → s
  33. Distance(TT, s) → aaa
  34. Circle(TT, 1.3) → q
  35. Line(Ε, Ζ) → a_1
  36. Intersect(s, a_1) → GG
  37. OrthogonalLine(GG, a_1) → b_1
  38. Mirror(s, b_1) → s'
  39. Segment(E, GG) → ggg
  40. If(y(GG) ≤ y(EE), t1 + ggg / abs(wt1b), 10) → t2
  41. Mirror(w, b_1) → w'
  42. Point(d_1) → Η
  43. Line(Η, a_1) → e_1
  44. Point(d_1) → Ι
  45. Circle(II, 1.3) → f_1
  46. Circle(III, 1.3) → g_1
  47. Circle(IIII, 1.3) → h_1
  48. Line(C, B) → i_1
  49. Intersect(i_1, a_1) → KK
  50. OrthogonalLine(KK, a_1) → j_1
  51. Mirror(i_1, j_1) → k_1
  52. Mirror(uu, j_1) → uu'
  53. Segment(C, KK) → aa
  54. If(β < 10° ∨ β > 350°, t1, t1') → t3
  55. Segment(KK, E) → LL
  56. Circle(KKK, abs(uu')) → p_1
  57. Intersect(p_1, k_1, 1) → Ν
  58. Line(KKK, E) → HH1
  59. OrthogonalLine(Ν, HH1) → m_1
  60. Intersect(HH1, m_1) → Ξ
  61. Vector(KKK, Ξ) → u_1
  62. OrthogonalLine(KKK, HH1) → n_1
  63. OrthogonalLine(Ν, n_1) → q_1
  64. Intersect(n_1, q_1) → Ο
  65. Angle(E, KK, KKK) → γ
  66. If(0 ≤ γ < 2° ∨ 360° ≥ γ > 358°, t3 + (LL - 0.8) / abs(uu'), If[γ ≥ 2° ∧ γ ≤ 3° ∨ γ ≥ 357° ∧ γ ≤ 358°, t3 + (LL - 0.7) / abs(uu'), t3 + (LL - 0.6) / abs(uu')]) → t4
  67. Angle(m_1, q_1) → δ
  68. Vector(KKK, Ο) → v_1
  69. Distance(E, k_1) → rrr
  70. Intersect(k_1, e_1) → Π
  71. Circle(Π, 2) → r_1
  72. Intersect(r_1, k_1) → I, J
  73. Vector(Π, I) → w_1
  74. Circle(IIIII, 1.3) → s_1
  75. Circle(IIIIII, 1.3) → t_1
  76. Circle(IIIIIII, 1.3) → c_2

AI Drawing Prompt Reference

Draw a billiard table as a polygon, marking the positions, speeds, and angles of the balls. Show the trajectories and collision effects of the balls as they move around the table, including reflections and changes in direction due to geometric transformations.
The prompt above was compiled from the original drawing commands and is for reference only.