The Heart Of Mathematics 2nd Pdf
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
where ![t in [-1,1]](https://mathworld.wolfram.com/images/equations/HeartCurve/Inline10.gif){kind=small}
(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](https://mathworld.wolfram.com/images/equations/HeartCurve/NumberedEquation3.gif){kind=small} | (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
A sixth heart curve is given by the simple expression
![x^2+[y-(x^2)^(1/3)]^2=1,](https://mathworld.wolfram.com/images/equations/HeartCurve/NumberedEquation5.gif){kind=small} | (9) |
(noted on a greeting card by J. Schroeder, pers. comm., Oct. 16, 2021). When properly nondimensionalized with scale paramaters 
and 
, the curve becomes
![(x/a)^2+[y/b-((x/a)^2)^(1/3)]^2=1,](https://mathworld.wolfram.com/images/equations/HeartCurve/NumberedEquation6.gif){kind=small} | (10) |
which can be written as a sextic equation in 
and 
.
The areas of these hearts are
 |  |  | (11) |
 |  |  | (12) |
 |  |  | (13) |
 |  |  | (14) |
 |  |  | (15) |
 |  |  | (16) |
 |  |  | (17) |
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.
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The Heart Of Mathematics 2nd Pdf
Source: https://mathworld.wolfram.com/HeartCurve.html
Posted by: lynnbereest.blogspot.com
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