Also named "snail curve
", the limaçon of Pascal is a two-dimensional parametric curve
defined by the equation
r = a + b sinφ
in polar coordinates. This curve is named after Etienne Pascal, father of Blaise Pascal. For values of b smaller than 1, the curve is generally called an ordinary limaçon. If b = 0, the curve is obviously a circle; if b = 1, the equation defines a cardioid, so that an ordinary limaçon looks like something halfway between a circle and a heart. For values of b greater than 1, a noose appears at the dip of the cardioid. The particular case of b = 2 is called trisectrix.
The limaçon of Pascal can be defined as an epitrochoid (or epicycloid) where both rolling and rolled circles are of the same diameter, or as a conchoid with a circle as for its generating curve.
You can observe a natural limaçon of Pascal when an internally reflective cylindrical surface is lit; for instance, when a filled mug is lit sideways, one often sees a limaçon of Pascal reflected on the liquid's surface. The same observation can be made on the bottom of a watch.
Primary source: www.2dcurves.com