Domain coloring

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Domain coloring plot of the function f(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]


Insufficient dimensions[edit]

A real function f : ℝ → ℝ (for example f(x) = x2) can be graphed using two Cartesian coordinates on a plane.

A graph of a complex function g : ℂ → 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[edit]

Given a complex number z = re, 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.

Unit circle domain coloring.png


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.

Sine Wave Domain Coloring Corrected.png

See also[edit]


  1. ^ Frank A. Farris, Visualizing complex-valued functions in the plane
  2. ^ Hans Lundmark (2004). "Visualizing complex analytic functions using domain coloring". Archived from the original on 2006-05-02. 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[edit]