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Comment by BMarkmann | Hacker News Reader

Wolfram Alpha does a pretty amazing job of taking an existing graphic and turning back into a graph-able equation... eg:

http://www.wolframalpha.com/input/?i=graph+looks+like+batman

http://www.wolframalpha.com/input/?i=graph+looks+like+bart+s...

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I feel like these are special cased, searching for: graph looks like <object> with any random object turns up junk.

Edit: then again it does work with Obama but not with Richard Nixon.

http://www.wolframalpha.com/input/?i=popular+curves

How can one come up with the exact equation to match such an exact drawing? i'm impressed. is it just tons of trial and errors?

Any continuous function can be described as a sum of sinus-functions with different amplitude and frequencies, given that the maximum frequency is limited. Finding these amplitudes can be done quite easily using something called discrete fourier transform. A lot of audio and imaging technology is based on this theory, jpeg compression for example, and almost any audio and signal-processing.

The downside is that very few mathematically interesting functions have limited frequency. While this is not a problem for signal processing, as our senses usually cannot sense the difference, in math it's very easy to see the difference even if you cut off at really high frequencies: http://upload.wikimedia.org/wikipedia/commons/d/d4/Synthesis...

In other words, if one tried to do the Batman curve using Fourier transform, the formula to get sub-pixel consistency would be extremely long.

This is easily solved in practice by segmentation. Images are segmented in blocks, and each small block can be well approximated by a low frequency content, but block boundaries can contain discontinuities. If you look at the wolfram alpha source, that's exactly how those images are constructed -- they are combinations low frequency segments, most likely obtained through fourier analysis.

Another trick used in frequency domain compression is they don't impose a hard cut-off of frequencies (truncation); instead they specify weights according to image quality, and lower are assigned to high frequencies, and those weights control the precision of each frequency.

All this works because real signals happen to be concentrated in certain frequencies, with a few discontinuities in between.

You don't need to do fourier transforms to do this. You can do it yourself and it is quite fun actually. It has been 13 years since I last did it but here is how you start:

1. Have a set of interesting graphs. sin(x), tan(x), log(x), e^x, 1/x, polynomials, hyperbola, sqrt(x)

2. Learn the effects of replacing x with f(x) -> sin(f(x)), log(f(x)), e^f(x). This is IMHO the most fun part. How will sin(1/x) look like? How will log(mod(sqrt(x))) look like?

3. Have a set of 'tricks'. What is the difference between sin(x) and sin(x) - A (ans. it moves the whole graph downward on y-scale). How to create mirror? (ans. take modulus). How to enlarge a graph? (ans. multiply with a constant).

4. Tackle the actual problem, edge by edge.

You can practice that by trying to hit your opponent in Graphwar: http://graphwar.com/play.html

How To Draw Einstein's Face Parametrically http://www.i-programmer.info/projects/119-graphics-and-games...

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