CARDIOID, a curve so named by G. F. M. M. Castillon (1708–1791), on account of its heart-like form (Gr. καρδία, heart). It was mathematically treated by Louis Carré in 1705 and Koersma in 1741. It is a particular form of the limaçon (q.v.) and is generated in the same way. It may be regarded as an epicycloid in which the rolling and fixed circles are equal in diameter, as the inverse of a parabola for its focus, or as the caustic produced by the reflection at a spherical surface of rays emanating from a point on the circumference. The polar equation to the cardioid is r = a(1 + cos θ). There is symmetry about the initial line and a cusp at the origin. The area is 32πa2, i.e. 112 times the area of the generating circle; the length of the curve is 8a. (For a figure see Limaçon.)