Domain coloring

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Domain coloring plot of the function f(x) =
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(x - 1)(x - 2 - i)/x + 2 + 2i, using the structured color function described below. Domain coloring x2-1 x-2-i x-2-i d x2+2+2i.xcf
Domain coloring plot of the function f(x) = (x − 1)(x − 2 − i)/x + 2 + 2i, using the structured color function described below.

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.

Contents

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.

Motivation

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.

Method

Complex coloring.jpg
Complex x hoch 3.jpg
HL plot of z, as per the simple color function example described in the text (left), and the graph of the complex function z3  1 (right) using the same color function, showing the three zeros as well as the negative real numbers as cyan rays starting at the zeros.

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.

Simple color function

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

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.

Discontinuous color changing

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.

Magnitude growth

A discontinuous color function. In the graph, each discontinuity occurs when
|
z
|
=
2
n
{\displaystyle |z|=2^{n}}
for integers n. Domain coloring z 03.jpg
A discontinuous color function. In the graph, each discontinuity occurs when for integers n.

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.

Highlighting properties

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]

History

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]

Limitations

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]

Related Research Articles

Complex number Element of a number system in which –1 has a square root

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 i2 = −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.

Exponential function Class of specific mathematical functions

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

Polynomial In mathematics, sum of products of variables, power of variables, and coefficients

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 x2 − 4x + 7. An example in three variables is x3 + 2xyz2yz + 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.

Hue Property of a color indicating balance of color perceived by the normal human eye

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.

Natural Color System

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.

Function (mathematics) Mapping that associates a single output value to each input

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.

Differentiable function Mathematical function whose derivative exists

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.

Graph coloring Assignment of colors to elements of a graph subject to certain constraints.

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 Perceived intensity of a specific 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.

Color scheme choice of colors used in design

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.

Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function. This article describes the classification of discontinuities in the simplest case of functions of a single real variable taking real values.

Tertiary color

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).

Heat map

A heat map is a data visualization technique that shows magnitude of a phenomenon as color in two dimensions. The variation in color may be by hue or intensity, giving obvious visual cues to the reader about how the phenomenon is clustered or varies over space. There are two fundamentally different categories of heat maps: the cluster heat map and the spatial heat map. In a cluster heat map, magnitudes are laid out into a matrix of fixed cell size whose rows and columns are discrete phenomena and categories, and the sorting of rows and columns is intentional and somewhat arbitrary, with the goal of suggesting clusters or portraying them as discovered via statistical analysis. The size of the cell is arbitrary but large enough to be clearly visible. By contrast, the position of a magnitude in a spatial heat map is forced by the location of the magnitude in that space, and there is no notion of cells; the phenomenon is considered to vary continuously.

Complex logarithm Logarithm of a complex number

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 1976L*, 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.

Color gradient specifies a range of position-dependent colors

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.

References

  1. May 2004. http://users.mai.liu.se/hanlu09/complex/domain_coloring.html Retrieved 13 December 2018.
  2. Poelke, Konstantin and Polthier, Konrad. https://web.archive.org/web/20181215222809/https://pdfs.semanticscholar.org/1b31/16583a2638f896d8e1dd5813cd97b3c7e2bd.pdf Retrieved 13 December 2018.
  3. 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)
  4. Frank A. Farris, Visualizing complex-valued functions in the plane
  5. 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.
  6. David A. Rabenhorst (1990). "A Color Gallery of Complex Functions". Pixel: The Magazine of Scientific Visualization. Pixel Communications. 1 (4): 42 et seq.
  7. 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)
  8. Douglas N. Arnold (2008). "Graphics for complex analysis" . Retrieved 2008-05-17.CS1 maint: discouraged parameter (link)
  9. "CET Perceptually Uniform Colour Maps". peterkovesi.com. Retrieved 2020-12-22.
  10. 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.
  11. 1 2 Kovesi, Peter (2017). "Colour Maps for the Colour Blind, presented at IAMG 2017" (PDF).