
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)
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A second heart curve is given by the parametric equations
(3)
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(4)
|
where
(H. Dascanio, pers. comm., June 21, 2003).
A third heart curve is given by
(5)
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(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)
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(8)
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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)
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(12)
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which arises through modifying the parametric equations of a nephroid (J. Mangaldan, pers. comm., Feb. 14, 2023).
The areas of these hearts are
(13)
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(14)
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(15)
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(16)
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(17)
| |||
(18)
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(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.