Also named "

snail curve", the limaçon of Pascal is a

two-dimensional parametric curve defined by the

equation

*r* = *a* + *b* sin*φ*

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