The astroid

Roll a circle of radius a/4 around the inside of a circle of radius a. A point on the small circle touches the big circle four times per trip, making four cusps. Curves made this way are hypocycloids; the astroid is the one with four cusps, and the deltoid has three.

Lean a ladder of length a against a wall and let it slide down until it lies on the floor. Every position of the ladder is tangent to the astroid x^(2/3) + y^(2/3) = a^(2/3): the astroid is the envelope of the falling ladder. Draw the line segments from (a cos t, 0) to (0, a sin t) for many values of t and the star appears.

The total length is 6a, and the area enclosed is 3πa²/8, three eighths of the circle around it. In the calculator, typing the implicit form with absolute values, abs(x)^(2/3) + abs(y)^(2/3) = 1, draws all four arcs; the parametric form draws the same curve.

Rolling a circle of radius a/k inside a circle of radius a gives a hypocycloid with k cusps: x = (a(k − 1)/k) cos t + (a/k) cos((k − 1)t), y = (a(k − 1)/k) sin t − (a/k) sin((k − 1)t). With k = 4 this simplifies to the astroid, a cos³t and a sin³t. With k = 3 it is the deltoid, a curved triangle; with k = 2 the rolling circle traces a straight line, a fact used in Cardan’s gear mechanism to turn rotation into straight-line motion.