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 and value represent the multiply-wrapped magnitude.

Domain coloring is a technique for visualizing functions of a complex variable. The term "domain coloring" was coined by Frank Farris, possibly around 1998.^[1][2] There were many earlier uses of color to visualize complex functions, typically mapping argument (phase) to hue.^[3] 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^[4] and colored grids were used in graphics by Doug Arnold that he dates to 1997.^[5] A comprehensive introduction to phase plots (a special version of domain coloring) is given in Elias Wegert's textbook.^[6]

Motivation

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 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 a three-dimensional space. One way of depicting holomorphic functions is with a Riemann surface.

Visual encoding of complex numbers

Given a complex number $z=re^{{i\theta }}$, the phase (also known as argument) $\theta$ 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 $-2\pi$ to $2\pi$ on the real axis and $-1.5$ to $1.5$ on the imaginary axis.

See also

• Bivariate map
• Color wheel graphs of complex functions

References

1. Frank A. Farris, Visualizing complex-valued functions in the plane
2. Hans Lundmark (2004). "Visualizing complex analytic functions using domain coloring". Retrieved 2006-05-25. Ludmark refers to Farris' coining the term "domain coloring" in this 2004 article.
3. David A. Rabenhorst (1990). "A Color Gallery of Complex Functions". Pixel: the magazine of scientific visualization. Pixel Communications. 1 (4): 42 et seq.
4. George Abdo & Paul Godfrey (1999). "Plotting functions of a complex variable: Table of Conformal Mappings Using Continuous Coloring". Retrieved 2008-05-17.
5. Douglas N. Arnold (2008). "Graphics for complex analysis". Retrieved 2008-05-17.
6. Elias Wegert (2012). Visual Complex Functions - An Introduction with Phase Portraits. Springer Basel. ISBN 9783034801799. Retrieved 6 January 2016.

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
• Domain Coloring Method on GPU
• Java domain coloring software (In development)
• MATLAB routines
• Python script for GIMP by Michael J. Gruber
• Matplotlib and MayaVi implementation of domain coloring by E. Petrisor
• MATLAB routines with user interface and various color schemes
• MATLAB routines for 3D-visualization of complex functions