TOPICS

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

 (1)

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

 (2)

A second heart curve is given by the parametric equations

 (3) (4)

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

A third heart curve is given by

 (5)

(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

 (6)

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

 (7) (8)

A sixth heart curve is given by the simple expression

 (9)

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

 (10)

which can be written as a sextic equation in and .

A seventh heart curve can be defined parametrically as

 (11) (12)

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

The areas of these hearts are

 (13) (14) (15) (16) (17) (18) (19) (20)

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.