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How do people come up with equations of curves to draw out complex objects?

Some popular examples would include: batman curve & PSY curve.

This stackexchange link explains the rationale for the batman curve nicely.

But other than trial and error, I can't see a reasonable way of drawing the much more complicated PSY curve.

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There seem to be $37$ people that Wolfram|Alpha has stored. –  Michael Albanese Jan 28 '13 at 6:11
See the corresponding question on the Mathematica StackExchange: mathematica.stackexchange.com/questions/17704/… (basically the same as @copper.hat's answer) –  Rahul Jan 28 '13 at 6:49
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1 Answer 1

Note that the Wolfram PSY curve is a parametric curve.

I would guess that the Wolfram PSY curve was created by drawing the curve first as a sequence of points in $\mathbb{R}^2$. This would correspond to a piece-wise affine ('linear') function $f:[0,1] \to \mathbb{R}^2$, with the property (among others) that $f(0)=f(1)$. Then take the Fourier series of $f$ and truncate at some point when the resulting curve looks reasonable.

This would be a straightforward (and tedious) way of drawing any 'closed' curve.

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So, it's kind of a pointillism artwork, followed by connecting the dots with piece-wise functions? –  Poseidonium Jan 28 '13 at 6:36
Hmm, I think embroidery would be a better metaphor given the underlying continuity; pointillism is more like pixelation in that it is essentially discrete... –  copper.hat Jan 28 '13 at 6:41
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