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Unwrapping Circles, Squares, and Equilateral Triangles

Unwrapping Circles, Squares, and Equilateral Triangles
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2015-07-16 21:07
CirclePolygonTransformationGeometric ConstructionLocus
① This example demonstrates what happens when you 'unwrap' a circle, a square, or an equilateral triangle into a straight line, visualizing the motion of boundary points and the resulting loci. ② Core mathematical concepts include circumference and π, perimeter of regular polygons, rotation, locus (trajectory), and the geometric idea of unwrapping, which relates to involute or logarithmic spiral curves. ③ Construction highlights: using GeoGebra commands such as Point, Circle, Polygon, Line, Centroid, Intersect, Rotate, Ray, CurveCartesian, and Locus, with objects like circle c, polygons poly1/poly2, points A through H, and loci loc1/loc2/loc3, created by rotating rays and tracing paths. ④ Suitable for high school or early college students, this interactive exploration bridges geometry and calculus, helping learners understand arc length, perimeter, and the connection between rolling/unwrapping and curve generation.

GeoGebra Commands & Parameters

  1. Point(yAxis) → A
  2. Circle(A, 1) → c
  3. Polygon((-1, -1), (1, -1), 4) → poly1
  4. Polygon((-1, -4), (1, -4), 3) → poly2
  5. Line(A, xAxis) → a
  6. Centroid(poly2) → B
  7. Line(B, xAxis) → b
  8. Intersect(c, a, 2) → C
  9. Intersect(xAxis, poly1) → D, E
  10. Intersect(b, poly2) → F, G
  11. Rotate(C, α, A) → C'
  12. Rotate(E, α, H) → E'
  13. Rotate(G, α, B) → G'
  14. Ray(H, E') → d
  15. Ray(B, G') → e
  16. Intersect(d, poly1) → I
  17. Intersect(e, poly2) → J
  18. CurveCartesian(t + 2, sin(t) + 3, t, 0, α) → j
  19. Locus(K, α) → loc1
  20. Locus(L, α) → loc2
  21. Locus(M, α) → loc3

AI Drawing Prompt Reference

Draw a coordinate plane with a unit circle centered at a point on the y-axis. Below it, place a square and an equilateral triangle, each with a horizontal base. Drop a perpendicular from the circle's center to the x-axis, and mark intersections: the circle with the perpendicular, the square with the x-axis, and the triangle with its median. Rotate these intersection points by the same angle around their respective reference points, then cast rays along the sides and trace where they meet the polygon edges. Also draw a sine curve as a reference. Finally, show the loci of the boundary points as they unwrap into straight lines, illustrating circumference and perimeter relations.
The prompt above was compiled from the original drawing commands and is for reference only.