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Thursday, January 31, 2013

Mathematical Heart


Valentines Day is coming up, and you less artistic folk might want to make a nice heart picture to send to your girlfriend. A parametrization of a heart shape is the following: x = 16\text{sin}^3(t)
y = 13\text{cos}(t) - 5\text{cos}(2t) - 2\text{cos}(3t) - \text{cos}(4t)
The heart is drawn as t ranges from 0 to 2\pi. How you increment t can alter the picture in cool ways. Interestingly, the area enclosed is exactly 180\pi


Blue, scale = \frac{3}{2},   t = 1, 2, 3, ...
Red, scale = 1,     t = \frac{1}{2}, \frac{2}{2}, \frac{3}{2}, ...
Black, scale = \frac{3}{5},   t = \frac{1}{100}, \frac{2}{100}, \frac{3}{100}, ...
Green, scale = \frac{1}{5},    t = k(\pi - .01), \; k = 0,1,2, ...



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