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Heart Curve


HeartCurves

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

 r(theta)=1-sintheta.
(1)

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

 (x^2+y^2-1)^3-x^2y^3=0.
(2)

A second heart curve is given by the parametric equations

x=sintcostln|t|
(3)
y=|t|^(0.3)(cost)^(1/2),
(4)

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

A third heart curve is given by

 x^2+[y-(2(x^2+|x|-6))/(3(x^2+|x|+2))]^2=36
(5)

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

HeartCurve5

A fourth curve is the polar curve

 r(theta)=2-2sintheta+sintheta(sqrt(|costheta|))/(sintheta+1.4)
(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

x=16sin^3t
(7)
y=13cost-5cos(2t)-2cos(3t)-cos(4t).
(8)

A sixth heart curve is given by the simple expression

 x^2+[y-(x^2)^(1/3)]^2=1,
(9)

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

 (x/a)^2+[y/b-((x/a)^2)^(1/3)]^2=1,
(10)

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

A seventh heart curve can be defined parametrically as

x=-sqrt(2)sin^3t
(11)
y=2cost-cos^2t-cos^3t,
(12)

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

The areas of these hearts are

A_0=3.661972725...
(13)
A_1=3/2pi
(14)
A_2=0.237153845...
(15)
A_3=36pi
(16)
A_4=12.52...
(17)
A_5=180pi
(18)
A_6=7.687...
(19)
A_7=(9pi)/(4sqrt(2)),
(20)

where A_4 can be given in closed form as a complicated combination of hypergeometric functions, inverse tangents, and gamma functions.

BonneProjection

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


See also

Bonne Projection, Cardioid, Heart Surface, Watt's Curve

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Cite this as:

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

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