A heart-shaped curve generated by a point of a circle that rolls (without slipping) on a fixed circle of the same diameter. In point-wise construction of the curve, let O be a fixed point of a circle C of diameter a, and Q a variable point of C. Lay off distance a along the secant OQ, in both directions from Q. The locus of the two points thus obtained is a cardioid (see illus.).If a rectangular coordinate system is chosen with O for origin initially and y axis tangent to C at O, the cardioid has the equation (x² + y² − ax)² = a²(x² + y²). The equation expressed in polar coordinates is ρ = a(1 + cos θ). The cardioid's area is 3/2πa², or six times the area of C, and its length is 8a.