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Rotation inside triangle

Rotation inside triangle
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2017-07-05 09:56
TriangleCirclePolygonCalculusTransformation
This example demonstrates the process of performing rotational transformations inside a triangle. Core mathematical concepts include rotation, integral calculus, arithmetic operations, and probability. The construction utilizes GeoGebra commands and objects such as CurveCartesian, Rotate, Circle, Polygon, Tangent, PolyLine, If, and Intersect. Suitable for high school mathematics students and teachers to explore geometric transformations and integral calculations.

GeoGebra Commands & Parameters

  1. CurveCartesian((((1 / 2 * r) * (k - 1)) * cos(t)) + ((1 / 2 * r) * cos(((k - 1) * t))), (((1 / 2 * r) * (k - 1)) * sin(t)) - ((1 / 2 * r) * sin(((k - 1) * t))), t, 0, (2 * π)) → a
  2. Rotate(a, π / 2, (0, 0)) → a'
  3. Circle((0, 0), 8) → c
  4. Rotate((0, 8), (2 * π / 3), (0, 0)) → A
  5. Rotate((0, 8), ((2 * 2) * π / 3), (0, 0)) → B
  6. Polygon(A, B, 3) → poly1, f, g, h, C
  7. Tangent(D, a') → i
  8. PolyLine(A, B, C, A) → j
  9. If(i ≟ i, i, yAxis) → l
  10. Intersect(l, j) → E, F
  11. If(0 ≤ Move ≤ (2 * π / 3), Polygon[E, C, A, F], If[(2 * π / 3) < Move ≤ ((2 * 2) * π / 3), Polygon[E, F, C, E], If[((2 * 2) * π / 3) < Move ≤ (2 * π), Polygon[E, B, C, F], If[(2 * π) < Move ≤ ((4 * 2) * π / 3), Polygon[E, B, F, E], If[((4 * 2) * π / 3) < Move ≤ ((5 * 2) * π / 3), Polygon[E, F, B, A], Polygon[F, E, A, F]]]]]) → poly2
  12. If(0 ≤ Move ≤ (2 * π / 3), Polygon[E, F, B, E], If[(2 * π / 3) < Move ≤ ((2 * 2) * π / 3), Polygon[E, A, B, F], If[((2 * 2) * π / 3) < Move ≤ (2 * π), Polygon[E, F, A, E], If[(2 * π) < Move ≤ ((4 * 2) * π / 3), Polygon[E, F, A, C], If[((4 * 2) * π / 3) < Move ≤ ((5 * 2) * π / 3), Polygon[E, F, C, E], Polygon[E, F, C, B]]]]]) → poly3

AI Drawing Prompt Reference

Draw a triangle with a circle centered at one of its vertices, where the circle's diameter equals the triangle's side length. Inside the triangle, plot a curve that results from rotational transformations. Use integral calculus and probability concepts to visualize the rotation process of the curve within the triangle, highlighting different sections of the curve with distinct colors based on their rotation angles.
The prompt above was compiled from the original drawing commands and is for reference only.