The Love Formula : How to Draw a Heart Shaped Curvy Graph
Ever wanted to know how to draw a heart shaped graph?
Is that real? well yes.
There are a number of mathematical curves that produced heart shapes, some of which are illustrated below.
Is that real? well yes.
The first curve is a rotated cardioid (whose name means "heart-shaped") given by the polar equation
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(1)
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The second is obtained by taking the 
cross section of the heart surface and relabeling the 
-coordinates as 
, giving the order-6algebraic equation
cross section of the heart surface and relabeling the -coordinates as , giving the order-6algebraic equation |
(2)
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The third curve is given by the parametric equations
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(3)
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(4)
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where ![t in [-1,1]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_upPs1Z3_w0TVkDhhhDYznksRlNqwr5DjRC3JoY9sTxLkfKGBEdAFa3dYAko-lQSmofMfWsXIIPUUVpFy0IIqXbLD4Cgy8c0CABsdwYpI_aYJilD6jppnpfNUhADrRbSOhKR1is3z7NUA=s0-d)
(H. Dascanio, pers. comm., June 21, 2003). The fourth curve is given by
(H. Dascanio, pers. comm., June 21, 2003). The fourth curve is given by![x^2+[y-(2(x^2+|x|-6))/(3(x^2+|x|+2))]^2=36](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_va4Q0_TrNBRO0gTm382h_XG-06QUpjejq9f--BQTZCINSRDh2su_Q702_njQuRYm7iRVFDJykhR_n3rZNjTFnRXSOjzwXxALkQJwijLxvnXBayigmwIEMKuvnPHfrWg5rMF9TSl3qi_TCuKddx4Ti09w=s0-d) |
(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. And the fifth curve is the polar curve
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(6)
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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 sixth heart curve can be defined parametrically as
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(7)
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(8)
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The areas of these hearts are
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(9)
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(10)
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(11)
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(12)
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(13)
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(14)
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where 
can be given in closed form as a complicated combination of hypergeometric functions, inverse tangents, and gamma functions.
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.
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