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Rotating and Scaling Circular Arc Parametric Curve Animation

Rotating and Scaling Circular Arc Parametric Curve Animation
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2016-09-16 05:47
TransformationParametric EquationGeometric Construction
In collections Geometry Review
This GeoGebra applet demonstrates a dynamic parametric curve animation generated through geometric transformations including rotation (Rotate) and dilation (Dilate). The core mathematical concepts cover geometric transformations, circle arcs and sectors, locus tracing, and parametric equations. Construction details: Draw base circles using the Circle command, determine arc endpoints via Intersect; apply Rotate for point rotation and Dilate for scaling transformations; construct circular arcs with CircleArc and sectors with CircleSector; use Min command to find extreme points and draw auxiliary rays; ultimately forming a parametric dynamic curve trajectory. This example is ideal for high school geometry transformation teaching and serves as an excellent visualization aid for learning parametric equations.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Point(c) → C
  3. Line(C, A) → a
  4. Intersect(c, a, 2) → D
  5. CircleArc(A, C, D) → d
  6. CircleArc(A, D, C) → e
  7. Point(e) → E
  8. Segment(C, D) → h
  9. Dilate(C, If[0 < b ≤ 1, 1 - b, 0], A) → C'
  10. Segment(C, C') → i
  11. Dilate(A, If[1 < b ≤ 2, 2 - b, 0], D) → A'
  12. Segment(A, A') → j
  13. Rotate(C, If[2 < b ≤ 3, ((-(b - 2)) * π), (-π)], A) → C'_1
  14. Rotate(C', If[2 < b ≤ 3, ((-(b - 2)) * π), (-π)], A) → C''
  15. Segment(C'_1, C'') → i'
  16. Rotate(C, If[3 < b ≤ 4, ((b - 3) * π), π], A) → C'_2
  17. CircleArc(A, C, C'_2) → k
  18. Dilate(D, If[5 < b ≤ 6, 6 - b, 0], E) → D'
  19. Segment(D, D') → f
  20. Ray(D, E) → g
  21. Point(g) → F
  22. Ray(E, F) → l
  23. Dilate(C, If[5 < b ≤ 6, 6 - b, 0], E) → C'_3
  24. Segment(C, C'_3) → m
  25. Min({m, f}) → n
  26. Circle(E, (0.2 * n)) → p
  27. Intersect(p, l) → G,
  28. Intersect(c, m, 1) → H
  29. Intersect(p, m) → I,
  30. CircleSector(E, G, I) → q
  31. Rotate(q, If[6.5 < b ≤ 7.5, ((b - 6.5) * π / 2), π / 2], E) → q'

AI Drawing Prompt Reference

Create an animation of a dynamic parametric curve showing geometric transformations like rotation and scaling. Start by drawing base circles and determining arc endpoints, then apply rotation and dilation transformations to construct circular arcs and sectors. Use the minimum command to find extreme points and draw auxiliary rays, ultimately forming a parametric dynamic curve trajectory for teaching geometric transformations and visualizing parametric equations.
The prompt above was compiled from the original drawing commands and is for reference only.