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A plane algebraic curve of order four which is described by a point $M$ of a circle of radius $r$ rolling on a circle with the same radius $r$; an epicycloid with modulus $m=1$. The equation of the cardioid in polar coordinates is:


In Cartesian coordinates it is:


The arc length from the cusp is:


The radius of curvature is:


The area bounded by the curve equals $S=6\pi r^2$. The length of the curve is $16r$. The cardioid is a conchoid of the circle, a special case of a Pascal limaçon and a sinusoidal spiral.

Figure: c020390a


[1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)
[a1] J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972)

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How to Cite This Entry:
Cardioid. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article