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Envelope of unit circles centered on an astroid

Envelope of unit circles centered on an astroid
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2020-01-28 16:44
CircleCoordinate GeometryOptimizationParametric EquationGeometric Construction
This example demonstrates the envelope formed by unit circles centered at points on an astroid. It involves core mathematical concepts like envelopes, parametric equations, coordinate geometry, and geometric constructions. The construction uses GeoGebra commands and objects such as Curve, Point, and Circle. Suitable for high school and above students to understand the geometric properties of parametric curves and envelopes.

GeoGebra Commands & Parameters

  1. CurveCartesian(cos(t)^(3), sin(t)^(3), t, 0, (2 * π)) → a
  2. Point(a) → A
  3. Circle(A, 1) → c
  4. CurveCartesian(cos(t)^(3) + sqrt(cos(t)^(6) + sin(t)^(6) - (3 * sin(t)^(4)) - (4 * cos(t)^(2)) + 3), ((cos(t) * (cos(t)^(3) + sqrt(cos(t)^(6) + sin(t)^(6) - (3 * sin(t)^(4)) - (4 * cos(t)^(2)) + 3))) + 1 - (2 * cos(t)^(2))) / sin(t), t, 0, (2 * π)) → b
  5. CurveCartesian(cos(t)^(3) - sqrt(cos(t)^(6) + sin(t)^(6) - (3 * sin(t)^(4)) - (4 * cos(t)^(2)) + 3), ((cos(t) * (cos(t)^(3) - sqrt(cos(t)^(6) + sin(t)^(6) - (3 * sin(t)^(4)) - (4 * cos(t)^(2)) + 3))) + 1 - (2 * cos(t)^(2))) / sin(t), t, 0, (2 * π)) → d