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滚圆螺旋线：直线与圆上的滚动圆绘制参数曲线

滚圆螺旋线：直线与圆上的滚动圆绘制参数曲线
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2019-05-30 23:09
圆变换参数方程滑块动画轨迹
① 本范例演示了当一个圆沿直线或另一个固定圆滚动时，其圆周上一点所描绘的轨迹，即经典的螺旋线（spirograph）图案。② 核心数学概念包括参数曲线、圆的旋转与平移、几何轨迹以及多尖点曲线的生成，涉及自然数（尖点数量）和旋转角度等参数。③ 作图要点：使用GeoGebra的Circle、Point、Rotate、Segment、Text、CurveCartesian、Line、Intersect等命令，构建了滚动圆、固定圆、旋转角度滑块、动画按钮和重置按钮等对象，通过参数控制滚动圆的半径、旋转角度和尖点数量，动态生成曲线。④ 适合中学高年级至大学阶段的数学教学，用于探索参数方程、几何变换和轨迹问题，也可作为数学艺术创作工具。

GeoGebra 命令及参数

  1. Circle(CenterCircleFixed, RadiusCircleFixed + RadiusCircleRollings) → c
  2. Circle(CenterCircleFixed, RadiusCircleFixed) → CircleFixed
  3. Point(CircleFixed) → D
  4. Rotate(D, β, CenterCircleFixed) → D'
  5. Point(xAxis) → A
  6. Point(xAxis) → R
  7. Segment(A, R) → f
  8. Text(If[AppearMachineryCircleBoolean, "☑ app. / disapp. machinery", "☐ appear / disappear machinery"]) → AppearMachineryTextCircle
  9. Text(If[AppearCurvesCircleBoolean, "☑ app. / disapp. curves", "☐ app. / disapp. curves"]) → AppearCurvesTextCircle
  10. CurveCartesian((RadiusCircleRolling * α) - (RadiusPen * sin(α)) + ScreenShiftLineX + 5, RadiusCircleRolling - (RadiusPen * cos(α)), α, 0, (4 * π)) → cycloid
  11. Circle(D', RadiusCircleRollings) → d
  12. CurveCartesian(((RadiusCircleFixed + RadiusCircleRollings) * cos(β)) + ScreenShiftCircleX + ScreenLeftX - (RadiusCircleRollings * cos(((RadiusCircleFixed + RadiusCircleRollings) / RadiusCircleRollings * β))), ((RadiusCircleFixed + RadiusCircleRollings) * sin(β)) - ScreenShiftCircleY - (RadiusCircleRollings * sin(((RadiusCircleFixed + RadiusCircleRollings) / RadiusCircleRollings * β))), β, 0, 360) → epicycloid
  13. CurveCartesian(((RadiusCircleFixed + RadiusCircleRollings) * cos(β)) + ScreenShiftCircleX + ScreenLeftX - (RadiusPens * cos(((RadiusCircleFixed + RadiusCircleRollings) / RadiusCircleRollings * β))), ((RadiusCircleFixed + RadiusCircleRollings) * sin(β)) - ScreenShiftCircleY - (RadiusPens * sin(((RadiusCircleFixed + RadiusCircleRollings) / RadiusCircleRollings * β))), β, 0, 360) → epitrochoid
  14. CurveCartesian(((RadiusCircleFixed - RadiusCircleRollings) * cos(β)) + ScreenShiftCircleX + ScreenLeftX + (RadiusCircleRollings * cos(((RadiusCircleFixed - RadiusCircleRollings) / RadiusCircleRollings * β))), ((RadiusCircleFixed - RadiusCircleRollings) * sin(β)) - ScreenShiftCircleY - (RadiusCircleRollings * sin(((RadiusCircleFixed - RadiusCircleRollings) / RadiusCircleRollings * β))), β, 0, 360) → hypocycloid
  15. CurveCartesian(((RadiusCircleFixed - RadiusCircleRollings) * cos(β)) + ScreenShiftCircleX + ScreenLeftX + (RadiusPens * cos(((RadiusCircleFixed - RadiusCircleRollings) / RadiusCircleRollings * β))), ((RadiusCircleFixed - RadiusCircleRollings) * sin(β)) - ScreenShiftCircleY - (RadiusPens * sin(((RadiusCircleFixed - RadiusCircleRollings) / RadiusCircleRollings * β))), β, 0, 360) → hypotrochoid
  16. Circle(CenterCircleRolling, RadiusCircleRolling) → CircleRolling
  17. Circle(CenterCircleRolling, RadiusPen) → CircleRollingPen
  18. Point(CircleRollingPen) → B
  19. Rotate(B, (-α), CenterCircleRolling) → Pen
  20. Segment(CenterCircleRolling, Pen) → PenArm
  21. Line(CenterCircleFixed, D') → PenArmsCircle
  22. Intersect(d, PenArmsCircle, 1) → CenterCircleRolling1
  23. Intersect(d, PenArmsCircle, 2) → CenterCircleRolling2
  24. Circle(CenterCircleRolling1, RadiusCircleRollings) → CircleRolling1
  25. Circle(CenterCircleRolling2, RadiusCircleRollings) → CircleRolling2
  26. Circle(CenterCircleRolling1, RadiusPens) → CircleRollingPen1
  27. Circle(CenterCircleRolling2, RadiusPens) → CircleRollingPen2
  28. Segment(CenterCircleFixed, CenterCircleRolling2) → MainArm
  29. Intersect(CircleRollingPen1, PenArmsCircle, 2) → ToRotatePen1
  30. Intersect(CircleRollingPen2, PenArmsCircle, 1) → ToRotatePen2
  31. Rotate(ToRotatePen1, ((-β) * RadiusRate), CenterCircleRolling1) → Pen1
  32. Rotate(ToRotatePen2, (β * RadiusRate), CenterCircleRolling2) → Pen2
  33. Segment(CenterCircleRolling1, Pen1) → PenArm1
  34. Segment(CenterCircleRolling2, Pen2) → PenArm2

AI 绘图提示语参考

画一个 Spirograph 图形，展示一个圆在直线或另一个固定圆上滚动时，其上的点所形成的轨迹。图形包含一个固定圆和一个可变半径的滚动圆，通过滑动条调整半径和旋转角度。展示滚动圆上的点在直线和圆上滚动时形成的参数曲线，包括内摆线、外摆线和内旋轮线。添加动画和重置按钮，动态显示轨迹的形成过程。
以上提示语为根据原始绘图命令整理而来，仅供参考。