This tool visualizes any complex-valued function as a conformal map by assigning a color to each point in the complex plane according to the function's value at that point.
Enter any expression in z.
The identity function z shows how colors are assigned: a gray ring at |z| = 1 and a black and white circle around any zero and colored circles around 1, i, -1, and -i. Checkers cover the plane in a 1/16th unit grid. Colors are turquoise in the positive direction, red in the negative, gold-green towards +i, purplish towards -i, and darker towards infinity. There is also a colored circle towards infinity at |z| > 16 that can be seen at any pole towards infinity such as in 1/z.
Here are some example functions to try:
Conformal maps have their history in 18th century mapmaking, when new mathematical developments allowed mapmakers to understand how to precisely eliminate local shape distortions in maps. Click the ⊕ button in the lower right corner to switch to a conformal mapping of the surface of the earth. Conformal maps preserve local angles everywhere, although they may distort sizes to do so.
The Mercator projection is an example. Try:
The azimuthal stereographic projection is a beautiful ancient technique that is also conformal, but it is usually broken into two hemispheres:
Lagrange advocated another conformal projection that squeezes the entire globe into a single circle:
To visualize the relationships within families of complex functions, parameterize them with the variables t, u, s, r, or n. The tool will render a range of complex functions for values of the parameter, adjustable with a slider or shown in an aimation. The parameter t will vary linearly from 0 to 1; u will circle through complex units; s follows a sine wave between -1 and 1; r follows a sine wave from 0 to 1 and back; and n counts integers from 1 to 60.
On the globe, multiplying by powers of unity will rotate the world on its axis:
Because more than 300 frames are computed, parameterized expressions can take a long time to fully render. A rough, blurry sketch is drawn quickly, and finer-grained rendering will follow for several minutes. When done, the frames will be antialiased and animated at 24 fps.
Simple families of rational function produce mesmerizing animations:
The radii of convergence can clearly be seen in the last two examplestool by David Bau