**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 32π*a*^{2}, *i.e.* 112 times the area of the generating circle; the length of the curve is 8*a*. (For a figure see Limaçon.)