I am not exactly sure if this question belongs here but I could not think of a better place to ask.

So I recently discovered that various people on the internet have created equations for rather complicated shapes such as the batman logo and have wondered how this was achieved, and whether such a procedure could be replicated on another image. Specifically, out of interest, I have wondered if it would be possible to generate an equation for the general shape depicted in the following image:


(full size)

The image itself is somewhat nonideal in that I would want an equation in this general shape, but symmetrical about an x-axis (aka a horizontal line through the center of the eye, also going through and bisecting the two fingers on each end). The four smaller "fingers" (two on each "hand") as well as the hairs/spikes on the center eye are not really that necessary if they make finding an equation too difficult. It is also ok if the equation merely yields the top half of the shape. Also important to note is that I don't care about the scaling of the axis or the equation (the equation could yield a figure 1 unit long, or one hundred units long)

That being said, I would like it if someone could point me to some program/software that would enable me to achieve this. Specifically, normal curve fitting and regression (at least on the level I can think of doing) seem non-applicable to this problem, and since the shape (even if just the top half) doesn't seem to be a function, the equation is probably a complicated implicit, polar, or parametric one, at which point my current level of math skills fail me. The best solution would be a program where I can input an image (such as an adjusted, more symmetrical version of the above) and it yields an equation. Less convenient, although still good, would be a program where I can input a set of "points" representative of the curve of this image and it yielding an equation. While it would also be nice if someone on here were to give the equation to me (especially if proprietary or premium software is involved), if one were to point me towards the software I should use for this then I think I would be able to take it from there. After doing my due diligence in research I have not been able to find one so I hope this community is the place to ask.

The context is that I desired the area of the figure for some calculations, but then realized the actual best-fit is probably to advanced for me to integrate and so I now desire the best-fit equation for personal curiosity. There may be a program for numerical integrating this figure, and while that would also be useful it would kind of defeat the point of wanting to take a look at the equation.

In summary, I want to find a way to generate an equation equivalent roughly to the shape in the image above or its top half.

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    $\begingroup$ There are detailed explanations in the discussion of the Batman image. $\endgroup$ – vadim123 Apr 28 '13 at 3:07
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    $\begingroup$ See Batman Curve as an example. This uses piecewise curve stiched together $\endgroup$ – Amzoti Apr 28 '13 at 3:09
  • $\begingroup$ What do all the colors mean? The Batman Curve is black and white, because it represents the points that do or do not satisfy an equation. One could use similar techniques to make an equation for the outer boundary of the figure. The basic shape reminds me of a Julia set which are often portrayed with colors for escape times. The proper Julia set is again black and white-each point is either in or out. $\endgroup$ – Ross Millikan Apr 28 '13 at 3:34

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