Limaçon Family

A limaçon, from the Old French for "snail", is a polar curve of the form r = a + b\cos(\theta). The curve was systematically studied by Étienne Pascal (father of Blaise Pascal) in the 17th century and characterized by Gilles de Roberval. The ratio a/b determines the shape: when a < b the curve has an inner loop, when a = b it becomes a cardioid, and when a > b it forms a dimpled or convex limaçon depending on how large a is. Limaçons appear in mechanical linkage design and as polar plots of certain antenna radiation patterns.

Here a single curve, r = a + \cos\theta (so b = 1), is drawn while the offset a animates from 0 to 2. At a < 1 the curve self-crosses into an inner loop; at a = 1 it closes into the cardioid; and as a grows past 1 it fills out from a dimpled to a smooth convex limaçon. The polar grid and equal aspect ratio preserve the radial geometry, so the whole one-parameter family reads off a single sweep.