In complex analysis, **domain coloring** or a **color wheel graph** is a technique for visualizing complex functions by assigning a color to each point of the complex plane. By assigning points on the complex plane to different colors and brightness, domain coloring allows for a four dimensional complex function to be easily represented and understood. This provides insight to the fluidity of complex functions and shows natural geometric extensions of real functions.

- Motivation
- Method
- Simple color function
- Discontinuous color changing
- History
- Limitations
- References
- External links

There are many different color functions used. A common practice is to represent the complex argument (also known as "phase" or "angle") with a hue following the color wheel, and the magnitude by other means, such as brightness or saturation.

A graph of a real function can be drawn in two dimensions because there are two represented variables, and . However, complex numbers are represented by two variables and therefore two dimensions; this means that representing a complex function (more precisely, a complex-valued function of one complex variable ) requires the visualization of four dimensions. One way to achieve that is with a Riemann surface, but another method is by domain coloring.

Representing a four dimensional complex mapping with only two variables is undesirable, as methods like projections can result in a loss of information. However, it is possible to add variables that keep the four dimensional process without requiring a visualization of four dimensions. In this case, the two added variables are visual inputs such as color and brightness because they are naturally two variables easily processed and distinguished by the human eye. This assignment is called a "color function". There are many different color functions used. A common practice is to represent the complex argument (also known as "phase" or "angle") with a hue following the color wheel, and the magnitude by other means, such as brightness or saturation.

The following example colors the origin in black, 1 in red, −1 in cyan, and a point at infinity in white:

There are a number of choices for the function . A desirable property is such that the inverse of a function is exactly as light as the original function is dark (and the other way around). Possible choices include

- and
- (with some parameter ).

A widespread choice which does not have this property is the function (with some parameter ) which for and is very close to .

This approach uses the HSL (hue, saturation, lightness) color model. Saturation is always set at the maximum of 100%. Vivid colors of the rainbow are rotating in a continuous way on the complex unit circle, so the sixth roots of unity (starting with 1) are: red, yellow, green, cyan, blue, and magenta. Magnitude is coded by intensity via a strictly monotonic continuous function.

Since the HSL color space is not perceptually uniform, one can see streaks of perceived brightness at yellow, cyan, and magenta (even though their absolute values are the same as red, green, and blue) and a halo around *L* = 1/2. Use of the Lab color space corrects this, making the images more accurate, but also makes them more drab/pastel.

Many color graphs have discontinuities, where instead of evenly changing brightness and color, it suddenly changes, even when the function itself is still smooth. This is done for a variety of reasons such as showing more detail or highlighting certain aspects of a function.

Unlike the finite range of the argument, the magnitude of a complex number can range from 0 to ∞. Therefore, in functions that have large ranges of magnitude, changes in magnitude can sometimes be hard to differentiate when a very large change is also pictured in the graph. This can be remedied with a discontinuous color function which shows a repeating brightness pattern for the magnitude based on a given equation. This allows smaller changes to be easily seen as well as larger changes that "discontinuously jump" to a higher magnitude. In the graph on the right, these discontinuities occur in circles around the center, and show a dimming of the graph that can then start becoming brighter again. A similar color function has been used for the graph on top of the article.

Equations that determine the discontinuities may be linear, such as for every integer magnitude, exponential equations such as every magnitude *n* where is an integer, or any other equation.

Discontinuities may be placed where outputs have a certain property to highlight which parts of the graph have that property. For instance, a graph may instead of showing the color cyan jump from green to blue. This causes a discontinuity that is easy to spot, and can highlight lines such as where the argument is zero.^{ [1] } Discontinuities may also affect large portions of a graph, such as a graph where the color wheel divides the graph into quadrants. In this way, it is easy to show where each quadrant ends up with relations to others.^{ [2] }

The method was probably first used in publication in the late 1980s by Larry Crone and Hans Lundmark.^{ [3] }

The term "domain coloring" was coined by Frank Farris, possibly around 1998.^{ [4] }^{ [5] } There were many earlier uses of color to visualize complex functions, typically mapping argument (phase) to hue.^{ [6] } 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^{ [7] } and colored grids were used in graphics by Doug Arnold that he dates to 1997.^{ [8] }

People who experience color blindness may have trouble interpreting such graphs when they are made with standard color maps.^{ [9] }^{ [10] } This issue can possibly be ameliorated by creating alternate versions using color maps that fit within the color space discernible to those with color blindness.^{ [11] } For example, for use by those with total deuteranopia, a color map based on blue/grey/yellow may be more readable than the conventional map based on blue/green/red.^{ [11] }

In mathematics, a **complex number** is a number that can be expressed in the form *a* + *bi*, where a and b are real numbers, and i is a symbol called the imaginary unit, and satisfying the equation *i*^{2} = −1. Because no "real" number satisfies this equation, i was called an imaginary number by René Descartes. For the complex number *a* + *bi*, a is called the **real part** and b is called the **imaginary part**. The set of complex numbers is denoted by either of the symbols or **C**. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.

In mathematics, an **exponential function** is a function of the form

In mathematics, a **polynomial** is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An example of a polynomial of a single indeterminate *x* is *x*^{2} − 4*x* + 7. An example in three variables is *x*^{3} + 2*xyz*^{2} − *yz* + 1.

In mathematics, a **singularity** is in general a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity.

In color theory, **hue** is one of the main properties of a color, defined technically in the CIECAM02 model as "the degree to which a stimulus can be described as similar to or different from stimuli that are described as red, orange, yellow, green, blue, purple," which in certain theories of color vision are called unique hues.

The **Natural Color System** (**NCS**) is a proprietary perceptual color model. It is based on the color opponency hypothesis of color vision, first proposed by German physiologist Ewald Hering. The current version of the NCS was developed by the Swedish Colour Centre Foundation, from 1964 onwards. The research team consisted of Anders Hård, Lars Sivik and Gunnar Tonnquist, who in 1997 received the AIC Judd award for their work. The system is based entirely on the phenomenology of human perception and not on color mixing. It is illustrated by a color atlas, marketed by NCS Colour AB in Stockholm.

In mathematics, a **function** is a binary relation between two sets that associates to each element of the first set exactly one element of the second set. Typical examples are functions from integers to integers, or from the real numbers to real numbers.

**HSL** and **HSV** are alternative representations of the RGB color model, designed in the 1970s by computer graphics researchers to more closely align with the way human vision perceives color-making attributes. In these models, colors of each *hue* are arranged in a radial slice, around a central axis of neutral colors which ranges from black at the bottom to white at the top.

In calculus, a **differentiable function** of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth and does not contain any break, angle, or cusp.

**Edge detection** includes a variety of mathematical methods that aim at identifying points in a digital image at which the image brightness changes sharply or, more formally, has discontinuities. The points at which image brightness changes sharply are typically organized into a set of curved line segments termed *edges*. The same problem of finding discontinuities in one-dimensional signals is known as step detection and the problem of finding signal discontinuities over time is known as change detection. Edge detection is a fundamental tool in image processing, machine vision and computer vision, particularly in the areas of feature detection and feature extraction.

In graph theory, **graph coloring** is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a **vertex coloring**. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a **face coloring** of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color.

**Colorfulness**, **chroma** and **saturation** are attributes of perceived color relating to chromatic intensity. As defined formally by the International Commission on Illumination (CIE) they respectively describe three different aspects of chromatic intensity, but the terms are often used loosely and interchangeably in contexts where these aspects are not clearly distinguished. The precise meanings of the terms vary by what other functions they are dependent on.

In color theory, a **color scheme** is the choice of colors used in various artistic and design contexts. For example, the "Achromatic" use of a white background with black text is an example of a basic and commonly default color scheme in web design.

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A **tertiary color** or **intermediate color** is a color made by mixing full saturation of one primary color with half saturation of another primary color and none of a third primary color, in a given color space such as RGB, CMYK or RYB (traditional).

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In the branch of mathematics known as complex analysis, a **complex logarithm** is an analogue for nonzero complex numbers of the logarithm of a positive real number. The term refers to one of the following:

In colorimetry, the **CIE 1976***L**, *u**, *v****color space**, commonly known by its abbreviation **CIELUV**, is a color space adopted by the International Commission on Illumination (CIE) in 1976, as a simple-to-compute transformation of the 1931 CIE XYZ color space, but which attempted perceptual uniformity. It is extensively used for applications such as computer graphics which deal with colored lights. Although additive mixtures of different colored lights will fall on a line in CIELUV's uniform chromaticity diagram, such additive mixtures will not, contrary to popular belief, fall along a line in the CIELUV color space unless the mixtures are constant in lightness.

In computer graphics, a **color gradient** specifies a range of position-dependent colors, usually used to fill a region. For example, many window managers allow the screen background to be specified as a gradient. The colors produced by a gradient vary continuously with position, producing smooth color transitions.

*Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.*

- ↑ May 2004. http://users.mai.liu.se/hanlu09/complex/domain_coloring.html Retrieved 13 December 2018.
- ↑ Poelke, Konstantin and Polthier, Konrad. https://web.archive.org/web/20181215222809/https://pdfs.semanticscholar.org/1b31/16583a2638f896d8e1dd5813cd97b3c7e2bd.pdf Retrieved 13 December 2018.
- ↑ Elias Wegert (2012).
*Visual Complex Functions: An Introduction with Phase Portraits*. Springer Basel. p. 29. ISBN 9783034801799 . Retrieved 6 January 2016.CS1 maint: discouraged parameter (link) - ↑ Frank A. Farris, Visualizing complex-valued functions in the plane
- ↑ Hans Lundmark (2004). "Visualizing complex analytic functions using domain coloring". Archived from the original on 2006-05-02. Retrieved 2006-05-25.CS1 maint: discouraged parameter (link) Ludmark refers to Farris' coining the term "domain coloring" in this 2004 article.
- ↑ David A. Rabenhorst (1990). "A Color Gallery of Complex Functions".
*Pixel: The Magazine of Scientific Visualization*. Pixel Communications.**1**(4): 42 et seq. - ↑ George Abdo & Paul Godfrey (1999). "Plotting functions of a complex variable: Table of Conformal Mappings Using Continuous Coloring" . Retrieved 2008-05-17.CS1 maint: discouraged parameter (link)
- ↑ Douglas N. Arnold (2008). "Graphics for complex analysis" . Retrieved 2008-05-17.CS1 maint: discouraged parameter (link)
- ↑ "CET Perceptually Uniform Colour Maps".
*peterkovesi.com*. Retrieved 2020-12-22. - ↑ Farris, Frank A. (2 June 2015).
*Creating Symmetry: The Artful Mathematics of Wallpaper Patterns*. Princeton University Press. pp. 36–37. ISBN 978-0-691-16173-0. - 1 2 Kovesi, Peter (2017). "Colour Maps for the Colour Blind, presented at IAMG 2017" (PDF).

Wikimedia Commons has media related to . Complex color plots |

- Samuel Li's function plotter
- 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
- Color wheel method
- Real-Time Zooming Math Engine
- Fractal Zoomer : Software that utilizes domain coloring

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