Blumenthal, Leonard M. Formerly, Department of Mathematics, University of Missouri, Columbia, Missouri.
Last reviewed:January 2020
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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 illustration). 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 , or six times the area of C, and its length is 8a. See also: Analytic geometry
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