Domain coloring

Domain coloring plot of the function
ƒ(x) =(x² − 1)(x − 2 − i)²/(x² + 2 + 2i). The hue represents the function argument, while the saturation represents the magnitude.

Domain coloring is a technique for visualizing functions of a complex variable. The term "domain coloring" was coined by Frank Farris [1] possibly around 1998. But the technique of using continuous color to map points from domain to codomain or image plane was used in 1999 by George Abdo and Paul Godfrey [2] and colored grids were used in graphics by Doug Arnold that he dates to 1997 [3].

Motivation

Insufficient dimensions

A real function $f:\mathbb{R}\rightarrow{}\mathbb{R}$ (for example f(x) = x²) can be graphed using two Cartesian coordinates on a plane.

A graph of a complex function $g:\mathbb{C}\rightarrow{}\mathbb{C}$ of one complex variable lives in a space with two complex dimensions. Since the complex plane itself is two dimensional, a graph of a complex function is an object in four real dimensions. That makes complex functions difficult to visualize in our three dimensional space. One way of depicting holomorphic functions is with a Riemann surface.

Visual encoding of complex numbers

Given a complex number z = re^iθ, the phase (also known as argument) θ can be represented by a hue, and the modulus r = | z | is represented by either intensity or variations in intensity. The arrangement of hues is arbitrary, but often it follows the color wheel. Sometimes the phase is represented by a specific gradient rather than hue.

Example

The following image depicts the sine function w = sin(z) from − 2π to 2π on the real axis and − 1.5 to 1.5 on the imaginary axis.

See also

• Conformal pictures where the domain is colored with a picture and not with a fixed color wheel.

References

http://www.ima.umn.edu/~arnold/complex.html

External links

• Color Graphs of Complex Functions
• Visualizing complex-valued functions in the plane.
• Gallery of Complex Functions
• Complex Mapper by Alessandro Rosa
• John Davis software - S-Lang script for Domain Coloring
• Open source C and Python domain coloring software
• Enhanced 3D Domain coloring
• Java domain coloring software (In development)