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滚动硬币问题：圆沿圆弧滚动时的旋转角与期望值

滚动硬币问题：圆沿圆弧滚动时的旋转角与期望值
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2020-05-05 22:00
圆角概率统计变换几何作图
① 本范例演示一个硬币（红色圆）沿另一条圆弧轨道滚动时，其自身旋转角度与滚动距离之间的关系，并探讨随机滚动后的期望旋转角。② 核心数学概念包括圆的滚动（无滑动滚动）、旋转角与弧长关系、圆周角与圆心角、余弦定理、期望值计算，以及一般四边形中的角度关系。③ 作图要点：使用GeoGebra的Rotate、OrthogonalLine、Angle、Intersect、Circle、Line、Segment、If、Point、CircleArc等命令，构造了红色圆、绿色圆、棕色圆、圆弧轨道、圆心连线、垂线、交点、弧上点、半圆等对象，并通过滑块或参数控制滚动距离，动态显示旋转角α、β、δ、ε、η等。④ 适合高中或大学低年级学生学习圆滚动几何、旋转与角度测量，以及概率期望在几何问题中的应用，也可用于数学建模或探究性学习。

GeoGebra 命令及参数

  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 绘图提示语参考

画一个红色圆圈沿着弧形轨道无滑动滚动，展示其旋转角度与滚动距离的关系，并探索随机滚动下的预期旋转角度。包含红色、绿色和棕色圆圈，弧形轨道，中心线，垂线，交点，弧上的点，半圆，以及用于控制滚动距离的动态滑块。显示角度α、β、δ、ε、η，并利用旋转、垂线、交点等几何变换构建图形。
以上提示语为根据原始绘图命令整理而来，仅供参考。