Heart Curve

There are a number of mathematical curves that produced heart shapes, some of which are illustrated above. A "zeroth" curve is a rotated cardioid (whose name means "heart-shaped") given by the polar equation

The first heart curve is obtained by taking the y=0 cross section of the heart surface and relabeling the z-coordinates as y, giving the order-6 algebraic equation

A second heart curve is given by the parametric equations

where t in [-1,1] (H. Dascanio, pers. comm., June 21, 2003).

A third 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.

A fourth 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.

A fifth heart curve can be defined parametrically as

A sixth 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 a and b, the curve becomes

which can be written as a sextic equation in x and y.

A seventh heart curve can be defined parametrically as

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

The areas of these hearts are

where A_4 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.

See also

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Cite this as:

Weisstein, Eric W. "Heart Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HeartCurve.html

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