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Small Circle Rolling on a Larger Circle

Small Circle Rolling on a Larger Circle
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2014-05-13 17:45
CircleTransformationGeometric Construction
In collections Circumference, Area, and Volume
This example demonstrates a small circle rolling along the circumference of a larger circle. Key mathematical concepts include circle properties, geometric transformations, and loci. It utilizes commands such as Point, Circle, Segment, Rotate, Intersect, Tangent, Mirror, Center, Vector, Translate, and CurveCartesian. Suitable for high school mathematics education and learning geometric transformations.

GeoGebra Commands & Parameters

  1. Point(yAxis) → A
  2. Circle(A, 4) → c
  3. Circle(A, 5) → d
  4. Point(d) → B
  5. Segment(B, A) → a
  6. Rotate(B, n, A) → B'
  7. Rotate(B, n, A) → N
  8. Circle(B', 1) → e
  9. Point(e) → C
  10. Rotate(C, 5 * n, B') → M
  11. Segment(A, M) → b
  12. Intersect(c, e) → G, , ,
  13. Tangent(G, c) → f,
  14. Mirror(e, f) → e'
  15. Center(e') → H
  16. Vector(C, N) → u
  17. Translate(H, u) → H'
  18. Rotate(H', -3 * n, H) → H''
  19. Segment(A, H'') → g
  20. CurveCartesian(3 * cos(t) + cos(-3 * t), 3 * sin(t) + sin(-3 * t), t, 0, 6.28319) → h
  21. CurveCartesian(5 * cos(t - π) + cos(5 * t), 5 * sin(t - π) + sin(5 * t), t, 0, 6.28319) → i

AI Drawing Prompt Reference

Draw a large circle with radius 5 and a small circle with radius 4 that is internally tangent to the large circle. Take a point B on the large circle and rotate B around A to get B'. Draw a circle e with radius 1 centered at B'. Take a point C on e and rotate it around B' to get M, then draw a line segment AM. At the intersection G of e and the large circle, draw a tangent line f. Reflect e over f to get e', and H is the center of e'. Vector u goes from C to N, and H' is H translated by u. Rotate H' counterclockwise by three times the rotation angle to get H'', then draw a line segment AH''. Finally, draw two curves: 3cos(t)+cos(-3t) and 3sin(t)+sin(-3t), and 5cos(t-π)+cos(5t) and 5sin(t-π)+sin(5t), with t ranging from 0 to 2π.
The prompt above was compiled from the original drawing commands and is for reference only.