February 10, 2013

Conformal Map Viewer

Have you ever wanted a visualization tool for complex functions?

While reading some complex number proofs I wanted one, but online I could only find installable software or Java applets (to be avoided because of security problems). So I wrote a Javascript conformal map viewer, which you can see here (click here to view it as a full page - it is about 1000 lines of javascript on a single page, and it is a nice example of doing canvas rendering with web workers):

The starting function "(z)" is the identity function, and shows how the tool colors the complex plane, with a ring at |z| = 1 and a small circle at |z| < 1/16, and 1/16th unit colored checkers on the rest of the plane. There is also a colored circle shown towards infinity, at |z| > 16. Colors are turquoise in the positive direction, red in the negative, gold-green in the "+i" direction, and they get darker as you go out towards infinity. The tool draws a quick fuzzy preview; wait a minute for it to complete the computation for a clear antialiased rendering. Lots of other functions can be typed into the box. For example, notice that |z|*e^(i*arg(z)) is the same as z.

Try visualizing the complex values in z^2, sin(z), e^z, log(z), sech(z), arctan(z), z^3-1, sin(z^3-1)/z, Jacobi elliptical functions sn(z,0.3), the Gamma function gamma(z), or a polynomial to squeeze a circle into a square 0.926(z+7.3857e-2 z^5+4.5458e-3 z^9).

Or try this: iter(z+z'^2,z,12). Recognize it?

Posted by David at February 10, 2013 01:54 AM
Comments

Thank you so much for your work! This is great!

Posted by: Peter Webb at March 8, 2013 05:34 PM

yes, thank you David. Beautifully executed!

The earth projection reminds me of an enjoyable article in "The Best Writing on Mathematics 2012" titled "Mathematics Meets Photography: The Viewable Sphere", a portion of which is available online at:

http://www.maa.org/publications/periodicals/math-horizons/mathematics-meets-photography

Given that article, a nice extension to this function viewer would be to allow pointing it at an arbitrary image. Input images could be expected in the equirectangular projection.

Posted by: Roice Nelson at September 10, 2013 01:33 PM

Thanks David, this is fantastic. I've always wanted to see (or make) something like this that used vector fields rather than colours.

Leon

Posted by: Leon at October 14, 2013 09:54 PM
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