Cardiod (Epicycloid, Hypocycloid) via rolling coin
Toggle "animate" to start the outer (planet) coin rolling/revolving round the inner (sun) coin. Toggle "trace" to leave a trail of the pink point, and "curve" to show the path that it travels. The "orbits" slider extends the trace/curve to cover multiple revolutions.
Notice that in addition to using the slider to set the "ratio" value, you may also type decimals or fractions into the number field.
A point on a circle rolling along the outside of a fixed circle traces out an "epicycloid." When ratio = 1, the epicycloid becomes the special case of a "cardioid."
If the rolling circle is "flipped" to roll inside the fixed circle, a "hypocycloid" is traced.