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Multiple Rotations: Exploring Composite Rotations and Loci

Multiple Rotations: Exploring Composite Rotations and Loci
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2016-04-15 05:41
AngleTransformationGeometric ConstructionSlider AnimationLocus
① This GeoGebra example demonstrates the visual effect of multiple successive rotations of points around a center, producing intricate patterns that illustrate composite rotational transformations. ② Core mathematical concepts include rotation (Rotate), dilation (Dilate), locus (Locus), and angle measurement, focusing on rotational symmetry and properties of composed transformations. ③ Construction highlights: using commands such as Intersect, Circle, Point, Angle, and Segment to build the base figure, then applying Rotate to points like B and A to generate series B1, A1 through A7, and using Locus to trace paths, with Element managing object sequences, resulting in a multi-rotation pattern. ④ Suitable for middle or high school geometry classes or math enrichment, helping students understand iterative rotation, locus generation, and transformation composition, while also appreciating aesthetic symmetry.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Circle(A, B) → c
  3. Point(c) → C
  4. Angle(B1, C1, D1) → E1
  5. Dilate(C1, k, D1) → F1
  6. Rotate(F1, (((-1)^(A1) * a) * E1), D1) → G1
  7. Dilate(C2, k, D2) → F2
  8. Angle(B2, C2, D2) → E2
  9. Rotate(F2, (((-1)^(A2) * a) * E2), D2) → G2
  10. Dilate(C3, k, D3) → F3
  11. Angle(B3, C3, D3) → E3
  12. Rotate(F3, (((-1)^(A3) * a) * E3), D3) → G3
  13. Dilate(C4, k, D4) → F4
  14. Angle(B4, C4, D4) → E4
  15. Rotate(F4, (((-1)^(A4) * a) * E4), D4) → G4
  16. Dilate(C5, k, D5) → F5
  17. Angle(B5, C5, D5) → E5
  18. Rotate(F5, (((-1)^(A5) * a) * E5), D5) → G5
  19. Dilate(C6, k, D6) → F6
  20. Angle(B6, C6, D6) → E6
  21. Rotate(F6, (((-1)^(A6) * a) * E6), D6) → G6
  22. Dilate(C7, k, D7) → F7
  23. Angle(B7, C7, D7) → E7
  24. Rotate(F7, (((-1)^(A7) * a) * E7), D7) → G7
  25. Segment(C1, D1) → H1
  26. Segment(C2, D2) → H2
  27. Segment(C3, D3) → H3
  28. Segment(C4, D4) → H4
  29. Segment(C5, D5) → H5
  30. Segment(C6, D6) → H6
  31. Segment(C7, D7) → H7
  32. Locus(D, C) → lieu1
  33. Element(liste1, a) → b
  34. Dilate(C8, k, D8) → F8
  35. Angle(B8, C8, D8) → E8
  36. Rotate(F8, (((-1)^(A8) * a) * E8), D8) → G8
  37. Segment(C8, D8) → H8