Heart Curve

The name "heart curve" is used for a number of mathematical curves that produce heart shapes. Three examples given by implicit equations are illustrated above.

The first implicit heart curve is obtained by taking the cross section of the heart surface and relabeling the -coordinates as , giving the order-6 algebraic equation

The second implicit heart curve is given by the simple expression

noted on a greeting card by J. Schroeder (pers. comm., Oct. 16, 2021). When properly nondimensionalized with scale parameters and , the curve becomes

which can be written as a sextic equation in and .

The third implicit heart curve is given by

(P. Kuriscak, pers. comm., Feb. 12, 2006). Each half of this heart curve is a portion of an algebraic curve of order 6.

Three heart curves given by parametric equations are illustrated above. The first parametric heart curve is defined by

The second parametric heart curve is given by

which arises through modifying the parametric equations of a nephroid (J. Mangaldan, pers. comm., Feb. 14, 2023).

The third parametric heart curve is given by

where (H. Dascanio, pers. comm., June 21, 2003).

Three heart curves given by polar equations are illustrated above. The first polar heart curve is a rotated cardioid (whose name means "heart-shaped") given by

The second polar heart curve is the polar curve

due to an anonymous source and obtained from the log files of Wolfram|Alpha in early February 2010. Each half of this heart curve is a portion of an algebraic curve of order 12, so the entire curve is a portion of an algebraic curve of order 24.

The third polar heart curve is given by the polar equation

with (Wu 2025). Its equivalent implicit equation is

which was given earlier by Nick (2014). The curve consists of the right half of the ellipse together with its reflection across the -axis.

The areas of these hearts are

where can be given in closed form as a complicated combination of hypergeometric functions, inverse tangents, and gamma functions.

The Bonne projection is a map projection that maps the surface of a sphere onto a heart-shaped region as illustrated above.