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Rolling Coin Problem: Rotation Angle and Expected Value of a Circle Rolling Alon

Rolling Coin Problem: Rotation Angle and Expected Value of a Circle Rolling Alon
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2020-05-05 22:00
CircleAngleProbability & StatisticsTransformationGeometric Construction
① This example demonstrates a coin (red circle) rolling without slipping along an arc-shaped track, showing how its own rotation angle relates to the rolled distance, and explores the expected rotation angle under random rolling. ② Core mathematical concepts include rolling of circles, relation between rotation angle and arc length, inscribed and central angles, cosine law, expected value, and angle relations in general quadrilaterals. ③ Construction highlights: using GeoGebra commands such as Rotate, OrthogonalLine, Angle, Intersect, Circle, Line, Segment, If, Point, CircleArc, the construction includes red, green, and brown circles, an arc track, center lines, perpendiculars, intersection points, points on arcs, semicircles, and dynamic sliders to control rolling distance, with angles α, β, δ, ε, η displayed. ④ Suitable for high school or early college students learning geometry of rolling circles, rotation and angle measurement, and probability expectation in geometric contexts, also useful for mathematical modeling and inquiry-based learning.

GeoGebra Commands & Parameters

  1. Rotate(ArcMiddle, (-ArcAngle), ArcCentre) → ArcRight
  2. Rotate(ArcMiddle, ArcAngle, ArcCentre) → ArcLeft
  3. OrthogonalLine(RedCentre, xAxis) → VLineThruRedCentre
  4. Rotate(PointOnStraight, (-DistanceRolled), RedCentre) → RedEdgePoint
  5. Angle(RedEdgePoint, RedCentre, PointOnStraight) → α
  6. OrthogonalLine(GreenCentre, xAxis) → VLineThruGreenCentre
  7. Intersect(GreenCircle, VLineThruGreenCentre, 1) → A
  8. Rotate(A, ((-DistanceRolled)) / r, GreenCentre) → A'
  9. Angle(A', GreenCentre, A) → β
  10. Circle(A', 1) → e
  11. Intersect(Arc, VLineThruGreenCentre, (6.283185307179586, -3)) → D
  12. Angle(ArcMiddle, ArcCentre, ArcLeft) → δ
  13. Angle(ArcRight, ArcCentre, ArcMiddle) → ε
  14. Rotate(ArcLeft, (-(Mod[DistanceRolled, ((2 * π) * r)] / ArcRadius)), ArcCentre) → PointOnArc
  15. Angle(PointOnArc, ArcCentre, ArcLeft) → ζ
  16. Line(PointOnArc, ArcCentre) → LineArcCentreBrownCircle
  17. Circle(PointOnArc, 1) → c
  18. Intersect(c, LineArcCentreBrownCircle, 1) → BrownCentre
  19. Circle(BrownCentre, 1) → BrownCircle
  20. Rotate(PointOnArc, (-DistanceRolled), BrownCentre) → BrownEdgePoint
  21. Angle(BrownEdgePoint, BrownCentre, PointOnArc) → η
  22. Line(ArcCentre, xAxis) → HLineThruArcCentre
  23. Segment(BrownCentre, BrownEdgePoint) → BrownCircleLine
  24. If(DistanceRolled ≥ ((2 * π) * r), DistanceRolled - ((2 * π) * r), DistanceRolled) → DistanceMod
  25. Rotate(RedEdgePoint, 135°, RedCentre) → RedEdgePoint'
  26. Angle(RedEdgePoint, RedCentre, RedEdgePoint') → γ
  27. Circle(RedCentre, 0.5) → RedInnerCircle
  28. Rotate(RedEdgePoint, 135°, RedCentre) → RedEdgePoint'_1
  29. Angle(RedEdgePoint, RedCentre, RedEdgePoint'_1) → θ
  30. Rotate(RedEdgePoint, (-(135°)), RedCentre) → RedEdgePoint'_2
  31. Angle(RedEdgePoint'_2, RedCentre, RedEdgePoint) → ι
  32. Rotate(RedEdgePoint, 45°, RedCentre) → RedEdgePoint'_3
  33. Angle(RedEdgePoint, RedCentre, RedEdgePoint'_3) → κ
  34. Line(RedCentre, RedEdgePoint'_3) → h
  35. OrthogonalLine(RedCentre, h) → j
  36. Intersect(RedInnerCircle, h, 1) → RedCircleLeftEye
  37. Intersect(RedInnerCircle, j, 2) → RedCircleRightEye
  38. Intersect(RedInnerCircle, j, 1) → RedCircleSmileLeft
  39. Point(h) → RedCircleSmileRight
  40. CircleArc(RedCentre, RedCircleSmileLeft, RedCircleSmileRight) → RedCircleSmile
  41. Segment(RedEdgePoint, RedCentre) → RedCircleLine
  42. Rotate(BrownEdgePoint, 45°, BrownCentre) → BrownEdgePoint'
  43. Angle(BrownEdgePoint, BrownCentre, BrownEdgePoint') → λ
  44. Line(BrownEdgePoint', BrownCentre) → k
  45. OrthogonalLine(BrownCentre, k) → l
  46. Circle(BrownCentre, 0.5) → BrownInnerCircle
  47. Intersect(BrownInnerCircle, k, 2) → BrownCircleLeftEye
  48. Intersect(BrownInnerCircle, l, 1) → BrownCircleRightEye
  49. Intersect(BrownInnerCircle, l, 2) → BrownCircleSmileLeft
  50. Point(k) → BrownCircleSmileRight
  51. CircleArc(BrownCentre, BrownCircleSmileLeft, BrownCircleSmileRight) → BrownCircleSmile

AI Drawing Prompt Reference

Draw a red circle rolling without slipping along an arc-shaped track to show the relation between its rotation angle and rolled distance, exploring the expected rotation angle under random rolling. Include red, green, and brown circles, an arc track, center lines, perpendiculars, intersection points, points on arcs, semicircles, and dynamic sliders for rolling distance control. Display angles α, β, δ, ε, η, using rotations, perpendiculars, intersections, and other geometric transformations.
The prompt above was compiled from the original drawing commands and is for reference only.