Limaçons as Envelopes of Circles
A limaçon can be defined as the envelope of the circles that pass through a fixed point P and have centers on a given circle . In this Demonstration, P is the origin. You can vary the center of the circle (shown in red) to see the envelope (shown in blue) change. Unchecking the "show limaçon" checkbox lets you picture the envelope on your own.
Snapshot 1: when P is at center of the circle , the envelope is a circle
Snapshot 2: when P is in the interior of but not at its center, the envelope is a dimpled limaçon
Snapshot 3: when P is on , the envelope is a cardioid
Snapshot 4: when P is outside of , the envelope is a limaçon with an interior loop