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Circle Unwrapping

Circle Unwrapping
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2016-07-12 13:39
CirclePolygonTransformationGeometric Construction
In collections Circles
This example demonstrates how a circle unfolds into a straight line, visually illustrating geometric transformations and the relationship between 3D and 2D representations. Key math concepts include geometric transformations, polygons, circles, and inequalities. Construction highlights the use of commands like Segment, Point, Sequence, Midpoint, and Polygon in GeoGebra. Suitable for high school students and above studying spatial geometry, transformations, and net development, as well as for educational demonstrations.

GeoGebra Commands & Parameters

  1. Segment(A, ((2 * π), 0)) → f
  2. Point(f) → C
  3. Sequence(Dilate[C, i / n, B], i, 1, n, 1) → list1
  4. Sequence(Dilate[C', i / n, B], i, 1, n, 1) → list2
  5. Sequence((x(Element[list1, i]), y(B)), i, 1, n) → list1A
  6. Sequence(Rotate[Element[list1, i], (180° * (π - x(C)) / π), Element[list1A, i]], i, 1, n) → list1'
  7. Sequence(CircleArc[Element[list1A, i], Element[list1, i], Element[list1', i]], i, 1, n) → list1_{arc}
  8. Sequence((x(Element[list2, i]), y(B)), i, 1, n) → list2A
  9. Sequence(Rotate[Element[list2, i], (-180° * (π - x(C)) / π), Element[list2A, i]], i, 1, n) → list2'
  10. Sequence(CircleArc[Element[list2A, i], Element[list2', i], Element[list2, i]], i, 1, n) → list2_{arc}
  11. Midpoint(C, C') → D
  12. Sequence(Segment[Element[list1, i], Element[list2, i]], i, 1, n - 1) → list3
  13. Segment(D, C) → g
  14. Segment(D, C') → h
  15. Segment(C, C') → i
  16. Segment(A, B) → j
  17. Polygon(C, B, C') → poly1, c', c, b
  18. Segment((x(C'), y(C') - a), (x(C'), y(C') + a)) → k
  19. Segment((x(D), y(D) - a), (x(D), y(D) + a)) → l
  20. Segment(A, ((2 * π), 0)) → m