Domain coloring

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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].

[edit] Motivation

[edit] Insufficient dimensions

A real function $f:\mathbb{R}\rightarrow{}\mathbb{R}$ (for example $f (x) = x 2$ ) 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 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.

[edit] Visual encoding of complex numbers

Given a complex number $z = r e i θ$ , the phase (also known as argument) $θ$ can be represented by 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.

[edit] Using the third dimension to represent modulus

3D Domain coloring plot of the function
ƒ(x) =(x² − 1)(x − 2 − i)²/(x² + 2 + 2i). The hue represents the function argument, height represents the unsigned magnitude.

It is possible to enhance 2D domain colored diagrams by using the third dimension for the modulus like this^[1]:

$\sgn(\Re(f(z))) \cdot |f(z)|$