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Hypotrochoid: Triangles and Squares

Hypotrochoid: Triangles and Squares
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2018-12-09 18:36
CirclePolygonTrigonometryParametric Equation
This example demonstrates the generation of a hypotrochoid, a complex geometric curve traced by a point on the circumference of a smaller circle rolling around the inside of a larger fixed circle. It involves core mathematical concepts such as circles, polygons, trigonometry, parametric equations, and geometric constructions. The construction uses GeoGebra commands like Circle, Ray, Intersect, Polygon, Segment, along with objects like buttons, radii, distances, and maximum angles. It is suitable for high school and above, focusing on parametric equations, geometric transformations, and locus drawing.

GeoGebra Commands & Parameters

  1. CurveCartesian(((R - r) * cos(t)) + (d * cos(((R / r - 1) * t))), ((R - r) * sin(t)) - (d * sin(((R / r - 1) * t))), t, 0, tmax) → Hypotrocoid
  2. Circle((0, 0), R) → CR
  3. Ray((0, 0), A) → l
  4. Circle(A, r) → Caux
  5. Intersect(Caux, l, 1) → B
  6. Circle(B, r) → Cr
  7. Polygon(P1, P2, P3) → Pol1, p3, p1, p2
  8. Polygon(P4, P5, P6) → Pol2, p6, p4, p5
  9. Polygon(P7, P8, P9) → Pol3, p9, p7, p8
  10. Polygon(P10, P11, P12) → Pol4, p12, p10, p11
  11. Polygon(P11, P2, 4) → Pol5, f, g, h, i, C, D
  12. Polygon(P5, P7, 4) → Pol6, j, k, m, n, E, F
  13. Polygon(P1, P6, 4) → Pol7, p, q, s, t, G, H
  14. Segment(B, P1) → Segd