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Three Bounces Billiard Problem: Geometric Construction and Trajectory Analysis

Three Bounces Billiard Problem: Geometric Construction and Trajectory Analysis
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2018-09-07 05:37
TransformationOptimizationGeometric ConstructionSlider AnimationLocus
In collections Symmetry Explorations
This example demonstrates the geometric phenomenon of a ball bouncing three times between reflective surfaces. The key mathematical concepts include reflection laws, trajectory optimization, and geometric transformations. Construction involves GeoGebra commands such as Line, OrthogonalLine, Mirror, and Intersect to create reflection points, trajectory lines, and bounce points. Suitable for high school students or math enthusiasts for learning geometric transformations and optimization problems.

GeoGebra Commands & Parameters

  1. Line(A, B) → f
  2. OrthogonalLine(A, f) → g
  3. Point(g) → C
  4. OrthogonalLine(C, g) → h
  5. OrthogonalLine(B, f) → i
  6. Intersect(h, i) → D
  7. Point(f) → Bounce1
  8. Mirror(Target, h) → Target'
  9. Mirror(Target', i) → Target''
  10. Mirror(Target'', f) → Target'''
  11. Point(h) → Bounce3
  12. Segment(Bounce3, Target) → j
  13. Segment(Bounce3, Target') → k
  14. Point(i) → Bounce2
  15. Segment(Bounce2, Bounce3) → l
  16. Mirror(Bounce2, i) → G'
  17. Mirror(Bounce3, i) → F'
  18. Segment(G', F') → l'
  19. Mirror(Bounce3, i) → F'_1
  20. Mirror(Target', i) → Target''_1
  21. Segment(F'_1, Target''_1) → k'
  22. Segment(Bounce1, Bounce2) → m
  23. Mirror(F'_1, f) → H
  24. Mirror(Target''_1, f) → I
  25. Segment(H, I) → k''
  26. Mirror(G', f) → G''
  27. Mirror(F', f) → F''
  28. Segment(G'', F'') → l''
  29. Mirror(Bounce1, f) → E'
  30. Mirror(Bounce2, f) → G'_1
  31. Segment(E', G'_1) → m'
  32. Segment(Cue, Bounce1) → n
  33. Segment(Cue, Target''') → p
  34. Segment(C, A) → q
  35. Segment(A, B) → r
  36. Segment(B, D) → s
  37. Segment(D, C) → t

AI Drawing Prompt Reference

This is a geometric figure showing a ball bouncing three times between reflective surfaces. The figure includes a ball, three reflective surfaces, three bounce points, and trajectory lines connecting these points. The construction order starts from an initial point, finding the first bounce point using reflection laws, then the second and third bounce points, and finally drawing trajectory lines connecting bounce points to the target. This figure helps understand geometric transformations and optimization problems.
The prompt above was compiled from the original drawing commands and is for reference only.