All colors of a color model that can actually be output by a coloring method are represented in the three-dimensional color space . Each coloring method has its own color space. A list of all color loci (singular: chromaticity ) of a color model is the color bodies .
All procedures and associated devices and materials that can produce color are called coloring methods. Such processes are printers , monitors , exposures , art prints , varnishing or manual paint application.
Colors, more precisely referred to as color valence in colorimetry, are based on color stimuli that differ in their spectral composition. Due to the need to be able to define these differences exactly, different color models were developed. The basis for this are Graßmann's laws . Each color can be defined by a color name (descriptive words), but also by the numerical color location. Depending on the color model, the color can be clearly described with three such variables according to brightness, saturation and hue, but also according to light / dark, red / green and yellow / blue values.
A color system is the system for arranging the color valences that produce colors in different ways: by “mixing” light as light colors or using colorants on a carrier material as body colors . Depending on the application, a different number of basic colors can be used: at least three, but also four or more coloring substances are used. However, these are no longer independent of one another.
The color system always only represents the basic principle of color mixing, never the technical implementation of the coloring method. This is most evident in the color white . Depending on the coloring method used, this can be glistening and dazzling, but also matt and dull. Both ways of representing white do not contradict the underlying color system.
The color model arises from the abstract color system mostly three-dimensional for the practical representation, which can be of different shapes. Within the models, all colors are assigned unique numerical values, the color locations.
If the color location of a color is changed with the help of software within the same color model, information can be lost. These quality losses also arise through the transmission path to the coloring method. This cannot be prevented, only reduced. Increasing the number of differentiation levels within the color model and appropriate color management are most effective. The color model used is the carrier of the sharpness information and so the model in turn influences the sharpness result in photography.
The color body is the geometric body with which the color model can be represented. For Philipp Otto Runge it was the sphere, Erwin Schrödinger suggested the color body according to Siegfried Rösch , Wilhelm Ostwald chose the double cone and Harald Küppers the rhombohedron . It is a systematic arrangement in the dense, continuous context of all color locations of the underlying color system.
The color locus ( plural : color loci , not ink type) is located as a point in or on the color bodies, and is described in its position in the color space with appropriate coordinates. This place represents the agreed color. Color locations can be continuously described in color spaces, their real presentation in color atlases, on the other hand, is naturally only possible discontinuously.
The color space of a coloring method includes, if possible, all colors that can be represented within the color model. When realizing the color representation, all coloring methods are inevitably lossy. Some colors have a defined color location, but cannot be represented with the available colorants. The colors that can be represented form a body within the color model , also known as the gamut . This body is called the color space; in the ideal case, the color space can cover the entire color model. Color spaces are used to visualize differences between an ideal state and the required reality.
- The amount of colors presented in a color space is the totality of all color stimuli that are perceived by the sense of sight . For example, the set of colors that are visible on a screen forms a color space, in this case the device color space of the screen with the coordinates R (ot), G (green) and B (lukewarm).
- Several (material) basic colors can also span the respectively defined and limited measurement space of these colors. Only those color valences that are defined as independent of one another according to Graßmann's laws may be selected as the basic colors of a color model .
- Colors are quantified by a color model . A color model is a coordinate system with base coordinates corresponding to the selected color valences. The numbers given are position vectors of the color model; they can be given in the form of a tuple (here 3-tuple).
- Color spaces are a necessary tool in the colorimetry , wherein the device-induced transformation of reproductions and Design ( Color Management different) and object colors teachings . As a “model of reality”, a color space is subject to the limits of its definition.
- The aim of the design of color spaces is to achieve a match with human color perception within the model limits . The input device and output device must be coordinated with one another. Improved coloring methods in turn create new requirements for color management.
Color vision can be described in three dimensions. This statement is based on Graßmann's 1st law , so that a color space (here as a space of colors) is three-dimensional. The reason for this is the stimulus intensity at the three color receptors . The color valence (colloquially “the color” ) is represented by three vector lengths to the color location, a point in the color space. Three-dimensionality had long been known to painters and was first described by Thomas Young with the three-color theory .
When reproducing body colors , a spectrally correct reproduction is hardly possible, since different materials or device systems or the different coloring methods hardly leave the same impressions, and the (real) color is influenced by ambient conditions. The phenomenon of metamerism describes how colors can arise in different ways from three basic colors. For practical purposes, colors with three basic valences can usually be represented with sufficient accuracy as long as the conditions are not changed too much. The frequent diversity of spectral compositions is for the individual to three perceiving Zapf values displayed .
Transfer from additive to subtractive
The self-luminous properties of the additive color mixture create a high contrast range . The "radiance" of this luminance not only conveys a high impression of sharpness , but also allows color representations that are only possible through additive color mixing.
In the case of subtractive color mixing ( body color ), because other primary colors are used, different colors can be displayed than in the case of additive color mixing. The color spaces of devices with additive and subtractive color mixing differ fundamentally. On the other hand, both also contain many colors that they can represent together. Because of these similarities, color separation is possible in the first place.
It becomes problematic when the coloring methods are no longer considered under standard conditions . The subtractive color mixture “lives” from reflected light, while the additive color mixture uses self-luminous colors. Both react differently to the change in ambient light - even the best color management cannot (yet) achieve anything here.
A common practical case for color separation is the conversion of RGB data (additive, e.g. from the screen) into the CMYK system for printing (subtractive). The transition from additive to subtractive mixing takes place via a simple transformation of the color spaces from device to device, since the non-linear mixing behavior of the printing pigments as well as the color of the paper (possibly with a color cast ) must be taken into account. Since the color coverage is not linear during printing, the color space conversion is considerably more difficult. This requires special color spaces (ICC) or LUT (look-up table) created for this purpose.
Another problem with this conversion is the use of different amounts of color, three or four colors, or more as when using spot colors .
Black is also usually used in printing for the following reasons (in addition, carbon black is an effective colorant):
- If black or dark gray is to be displayed subtractively with a coloring method , it is more economical to use black as a separate color. The representation of black from only three colors is very complex, expensive, sometimes impossible due to the actual absorption of the color pigments and is therefore (almost) only used in color photography.
- The subtractive color mixture lacks the high contrast range that is characteristic of the additive mixture. The addition of black improves the subjective impression of contrast (the printer speaks of depth).
- Since printing processes are raster-oriented processes, there are strong subjective sharpness losses when displaying delicate colors. The grid width increases, so the image detail contains less information, which is interpreted by the eye as a loss of sharpness. The addition of black creates a subjective compensation for this loss. In contrast, for the same reason, gray values to be printed can often be better generated from composite colors than from black.
Due to problems such as non-linear color behavior, differences in the amount of color, loss of luminance and subjective sharpness compensation, color separation is very time-consuming.
In the case of photo reproduction , exposure is a clear advantage, as it uses the same color model (namely RGB ) as the input devices ( scanner , camera ) and the control device ( screen ). Only for the finished photo (coloring method) does the additive need to be transferred to the subtractive color mixture.
Color spacing and equidistance
There are no devices that can capture or generate the full color gamut of human perception. MacAdam was working on a colorimetry that should enable the equidistance of color distances, the color distances should be perceived as visually the same. Such colorimetry has the consequence that the parameters for the color differences depend on the position in the chromaticity diagram or chromaticity diagram.
Representing the human perception of color distances in technically defined color spaces results in tolerance ellipses of the same color perception in the CIE color space, known as MacAdam ellipses . Here is the starting point for the further development of higher colorimetry. Further work in this area was done by Walter S. Stiles and D. Farnsworth. Stiles developed a line element that describes equidistant perceived color distances mathematically as equidistant (with the same distance). Farnsworth developed a nonlinear transformation that deforms all MacAdams ellipses into circles. As a solution, the CIE initially created the UCS color space in several versions. Later (1976) both the Lab color space (preferred for body colors) and the LUV color space (preferred for light colors) were presented as equally spaced color spaces.
Although Leonardo da Vinci had already tried to arrange colors artistically, the attempts got stuck because of the lack of theoretical foundations. Around 1800, at the time of Goethe's interest in color theory, ideas about colors were very subjective. The goal was still primarily to make the relationships between colors easier for painters. Runge's color ball is an example .
Around 1900, advancing industrialization required numerical color specifications, and a design should be possible even without a color template currently available. The work of Munsell , Ostwald , Rösch , Schrödinger preceded this goal of bringing order to the variety of color nuances . Important physical principles come from Maxwell , Young , Hering . Measurements of color stimulus were carried out by William David Wright and Guild in 1928.
As a result of this work, the first standardization of a color space by the International Commission on Illumination (CIE) became possible. Elaborations of the CIE are recommendations which enable worldwide coordination of the device classes by the special committees.
The first color model was proposed by the CIE in 1931 with the tristimulus model . This model was based on the mean 2 ° normal observer (from a group of 17 test persons). This 2 ° field of view corresponds to the size of the retinal region with the densest packing of cones (color receptors) in the human eye, the fovea . Since the sample areas for sampling were larger, however, the tristimulus model for the 10 ° normal observer was introduced in 1964. Since today's dimensions for small color screens, for example for MP3 players, portable game consoles and cell phones, are very small, the 2 ° normal observer from 1931 is becoming increasingly important for small viewing angles. As early as the 1940s, MacAdam discovered a problem in the xy area: the perceptual unevenness in the XYZ model (also known as (shoe) sole) led to the xy area being transformed into the UCS system (Uniform Chromaticity Scale, Yuv and Yu'v ') was deformed in such a way that the color differences came very close to the ideal of perceptual uniformity (equality of color differences in color space and perceived color differences). In the original xy plane, the size of the tolerance ellipses fluctuates approximately by a factor of 20, with the smallest ellipses in the blue area and the largest ellipses in the green area of the diagram. In the UCS system of 1976 this non-uniformity was greatly reduced. The size of the tolerance ellipses in the CIE 1976 UCS diagram (u'v'-diagram) only fluctuates approximately by a factor of 4. According to MacAdam, this is the best value that can be achieved by transformations of this type.
The chromaticity area eliminates the third axis of the lightness reference value A, which is equated with the tristimulus value Y. The luminance value is also referred to as L (Luminance).
In 1976, the CIE adopted both the L * a * b * and the L * u * v * models. In both systems, the adjustment of the color differences in the color space to the perception is achieved in that both systems use a term for L * which contains the third root of the quotient of the tristimulus value Y and the white point Y n . This term is used to mimic the logarithmic brightness perception of the human visual apparatus . This non-linearity also flows into the values a * and b * or u * and v *. The non-linear transformation is reversible. The L * a * b * model is preferred for body colors and can be represented in polar coordinates (more precisely cylinder coordinates ), in the form of the L * C * h ° system, instead of in Cartesian coordinates . The cylindrical representation gives the additional coordinates C * (chroma) and the hue angle h ° (hue). The L * u * v * system is more suitable for light colors, as it has an associated chromaticity diagram. L * u * v * can also be converted into cylindrical coordinates with the additional parameters C * (chroma), h uv (hue). A third parameter, s uv (psychometric saturation), in contrast to the L * C * h ° system, can also be derived.
The development and standardization of photographic and electronic devices resulted in a number of specially selected RGB color spaces (sRGB, Adobe RGB 1998) that were adapted to the phosphors used for red, green and blue and to realizable filters (TFT screens) were. The aim is to do justice to the color stimuli that can be represented with it. In representations on the chromacity diagram (xy area of the CIE), RGB systems are color areas within the phosphors (material realizations of the radiation excited by electrons in the required spectral range). Since the xy-area (shoe sole, horseshoe en: horseshoe) defines the maximum perceptible colors, the RGB color types must lie within the spectral color range.
With the advancement of mathematical topology and , on the other hand, the increasing demands on the reproducibility of the color impression in electronic recording and playback technology, further adjustments to reality will be necessary. This trend is illustrated by the color distance formulas (ΔE), which define the dimension in the color space and were modified in 1976, 1994 and 2000. The ICC profiles represent a similar trend , with these application-oriented, also device-oriented working color spaces. In color management it is possible to determine the special color spaces of the devices for the adaptation of the color rendering / conversion with different device categories. Using matrix calculations or LUT (look-up tables), the color location is transformed from the special working space of the source device into a suitable (if possible) comprehensive color space as an intermediate result, in order to determine the color location in the working color space of the target device from this "intermediate space" (communication color space) .
Up to now 30 to 40 color models have been created, which differ in the intended area of application. Accordingly, they can be categorized into:
- Technical-physical models that generate the color stimulus from real or idealized coloring substances, such as
- Perceptual models that describe colors through the characteristics of brightness , saturation and hue , such as
- Special models for the color display of self-illuminating devices such as
Color distance formulas
Color differences can be quantified using color difference formulas . The result of such a formula, ΔE , is considered to be a fairly reliable indicator of perceived color differences. The color difference formula ΔE 1976, which has changed since the introduction of the Lab color space in 1976, and the development of its successors make it clear that this is by no means a trivial problem. ΔE 1976 was determined from the Euclidean distance measure between the color points. This simple calculation has been significantly further developed and expanded to CIE94 (ΔE 1994), and published in 1995. CIE94 was expanded again in 2000 to CIEDE2000 (ΔE 2000). Strictly speaking, CIEDE2000 is a hybrid model, since not only the color distance formulas have been changed, but a simple transformation of the LAB color space also precedes the actual color distance calculation. The way of adapting the color space was completely implemented in the DIN99 color space. The color difference formula remains unaffected and is identical in structure to the original ΔE 1976. Another common color difference formula is AE CMC (l: c), developed by the C olour M easurement C ommittee of the Society of Dyers and Colourists of Great Britain (color measurement committee of the Society of Dyers and Colorists Britain), which was published 1984th
In the further development, earmarked factors were introduced early on. Especially for the textile industry (ΔE CMC (l: c)) special correction factors have been introduced into the calculations of the color difference. These factors can also be adjusted to determine the color distance in graphic applications.
Variation of the color spaces
The DIN99 color space occupies a special position. It was first published in 1999 as a color space according to DIN 6176 and later developed further to DIN 6176: 2001-03. Instead of adapting the color distance formulas, a complete transformation of the CIELAB color space was carried out. This allows color distances to be determined as Euclidean distances according to the same principle as ΔE of the CIELAB color space.
Economic importance of the color difference
The color difference is of interest for drafting contracts (which color must “Ferrari red” car paint be?) And also for color formulation. Especially with colors with a high recognition value , as is common with many brands as corporate identity , a consistently flawless color (re) production and rendering is very important. In the field of transport, colors for light signals such as traffic lights are precisely prescribed. Accordingly, they must be supplied by the manufacturer in the exact color. With "white-gray" (almost achromatic) colors, there is also the problem that even the smallest deviations can lead to clearly perceptible color casts (the colors of pants and jacket "bite"), which in many areas, for example when purchasing wall paints Patterns of clothing or car paint is not acceptable. There are serious economic consequences for the manufacturer or supplier.
Color samples and color catalog
The representation of color spaces is often realized through abstract topographical descriptions. An alternative to this are color samples in a color catalog . However, for technical reasons, only selected colors are presented. For all colors of a color space, i.e. the continuous transition of all color locations, this is conceivable for types of light, but not practically possible.
Variants of the color space shape
A color space describes the colors that can be recognized or displayed by an input device ( visual sense , camera , scanner ) or an output device ( screen , imager , printer , projector ) under specific conditions. Just as every person perceives colors individually, devices, at least device classes, also have different color spaces in which they register or display colors. Such individuality is due to production fluctuations and construction differences.
Further deviations are caused by optical effects that are not taken into account when measuring color spaces :
- Input devices ( sense of sight , camera , scanner ) change their color sensitivity to a great extent when there are significant differences in brightness . Since this case is more the rule than the exception in practice, a color space created under standard conditions can only be used as a guide.
- Output devices ( screen , imagesetter , printer , projector ) work under certain lighting conditions. Depending on the color temperature of the ambient light, the colors are perceived differently by the eye. Only an output device that is used under standard lighting conditions delivers results that come close to the previously determined color space.
A large part of these differences is corrected by automatic image optimization . In doing so, metameric effects are used which - to put it simply - simulate colors . This color simulation is technically highly developed and an integral part of everyday life. A typical example are inkjet printers which use a high proportion of black to hide the defects in the representation of colors.
Some color spaces and color models
Specialized models and their spaces play a role in many areas of application:
- LMS color space - the physiological color space that is based on the spectral sensitivities of the L, M, S cones.
- XYZ color space - the standard color space originally established by the CIE, constructed on computational coordinates X, Y, Z, which are created from cone sensitivities.
- RGB color space - computer monitors , Internet standard.
- CMYK color model - desktop publishing , final printing .
- HSV color space with the variants HSL, HSB, HSI - design , documentation of painting , video art .
- Lab color space - CIE color space derived from XYZ, which also includes all perceptible colors; and its further development DIN99 color space .
- LCh ° color space does not designate a further color space in the actual sense, but the representation of HSV, LUV or LAB in polar coordinates.
- I1I2I3 color space - computationally optimized space for image processing .
- YCbCr color model (sometimes called YCC for short, see below) - digital television, both digital PAL and digital NTSC , DVB , JPEG , MPEG , DVD-Video .
- xvYCC - compared to YCbCr extended color space that uses the entire 8 bits per color channel and can be used for new flat screens.
- YPbPr color model - analog HDTV , analog component video .
- YUV color model - for analog PAL and NTSC .
- YIQ color model - obsolete, previously used with analog NTSC .
- YDbDr color model - for analog SECAM .
- YCC color model - Kodak Photo CD.
Color values in the presentation of Internet pages
The specification of the color values in the Cascading Style Sheets is a clear example of a three-dimensional color model. The values are defined in the system of an RGB color model with red, green and blue. The application color space for CRT monitors is the color space of the screen with the typical phosphors that phosphoresce in red, green and blue when electrons are excited. The underlying standard is sRGB , which uses the color coordinates specified in the ITU-R BT.709-5 standard as primary valences.
In the "CSS (-rgb) model", values between 0 and 255 are defined for the background of the field of the website (ie in 2 8 values). The example says
- a "purple" with the following color values
- r = 255 for the ideal basic color red in full purity and strength,
- g = 0 accordingly for missing basic color green,
- b = 229 specifies that the basic color blue should be 1.055 × (229/255) 2.4 - 0.055 = 76% proportional strength (see sRGB color space )
- a "green" with the following color values, the (RGB) complementary color
- r = 0 missing base color red,
- g = 255 basic color green in full strength and intensity,
- b = 150 complementary basic color blue in 1.055 × (150/255) 2.4 - 0.055 = 24% proportional strength (24% + 76% = 100%)
The color values of the CSS are converted by the software used on the PC. The three screen luminescent substances for red, green and blue are controlled in terms of beam intensity. At a sufficient distance from the screen, this color stimulus leads to a color valence in the user's "individual cone color space ". This creates the desired color impression when looking at the website. The cone color space of the observer (“the one” who is “now” looking at “this” screen.) Is an LMS color space of the observer “individuums”.
Color system for screens
When mixing three primary colors ( RGB system ), colors can only be generated with conventional display devices, such as tube and LC screens, within the scope of the emission sources or by absorbing colorants (filters). Color systems with the same wavelength of color and lightness ( HSV ) are better suited to denote the pure colors; the technical interpretation is more difficult. The special position of the purple / magenta colors can be recognized in the horseshoe-shaped CIExy or CIELuv color diagram by the straight line that connects the outermost blue value with the outermost red value.
The CIE systems
- Tristimulus space from 1931 (2 ° normal observer), added in 1964 with new data sets for a field of view of 10 ° (10 ° normal observer)
CIE XYZ color space system ( chromaticity diagram )
- Standard color chart
- Since the spectral value function y (λ) corresponds exactly to the sensitivity to brightness when viewing the cones, the (non-normalized) coordinate Y can be used as the brightness value (luminance); it is better to choose “A” instead of Y as a constant
- Coordinates: Y, x, y or only x, y in the standard color table
CIEYUV color space system , also CIE UCS, proposed in 1960, (CIE 1960 UCS, Uniform Chromaticity Scale, de: Einheitliche Farbskala )
- linear transformation by CIEXYZ in order to reduce the location-dependent non-linearity of the perceived color differences
- only defined 2D color distances
- Despite often the same spelling (YUV instead of CIEYUV or CIEYuv) not related to YUV from video technology!
- Coordinates: Y, u, v
YUV color model , (CIE 1964 UCS), proposed in 1964
- linear transformation of the CIEYUV (Yuv) color space
- further linear transformation in order to reduce the location-dependent non-linearity of the perceived color differences
- only defined 2D color distances
- Coordinates: Y, u ', v'
CIELUV color space system from 1976
- non-linear transformation of the CIEXYZ color space, includes the CIE 1964 UCS color space (CIEYU'V ') for the white point, transformation is reversible
- The spectral line is the outer limit of the color diagram, so the absolute saturation (relative to the spectral line) of a color can be measured
- Color mixtures lie on straight lines in space, therefore very well suited for colorimetric calculations and the representation of additive color mixing, such as for describing self-illuminating sources (light colors)
- Color diagram oriented psychometrically and based on the opposite color theory, similar to CIELAB
- Focus on equidistant color perception, comparable to the LAB color space
- Coordinates: L *, u *, v *
CIELAB color space system from 1976
- non-linear transformation from CIEXYZ, transformation is reversible
- Color diagram oriented psychometrically and based on the opposite color theory
- In contrast to CIELUV, in CIELAB it is not possible to measure the absolute saturation (relative to the spectral line) because the spectral line does not have a preferred position in the LAB color space and the chroma of a color is shown in the diagram instead of the saturation
- no color plate available
- good equidistance of color distances guaranteed by the nonlinear transformation
- especially used to describe non-luminous colors (body colors)
- Coordinates: L *, a *, b *
Systems outside the CIE
DIN99 color space , (color space according to DIN 6176), since 1999
- Coordinates: , ,
- Alternative to the color distance formula CIE94, comparable equidistant
- Later further development (DIN 6176: 2001-03) improved equidistance. The currently best variant (DIN99d) is qualitatively between CIE94 and CIEDE2000
- Special position compared to all CIELAB successors, instead of the color difference formulas, the entire color space is transformed
- Adjustment of , dark colors weighted more heavily, light colors compressed
- "Round" shape (equidistant ideal shape) through radial compression of the hue plane, thereby:
- Color differences near the achromatic axis are weighted more heavily through the radial compression of saturated colors (similar to CIEDE2000)
- Calculation easier than CIE94 and CIEDE2000, is calculated as, because the color space is transformed and the color distance formulas remain unaffected
- In the case of the DIN99d variant, part of the transformation already takes place in the XYZ color space
Color components that are essential for the CIE color space systems:
Color / Image Appearance Models (CAM / IAM)
At the moment, intensive investigations and research are being carried out in the field of “Color Appearance Models” ( CAM ), in German: models for the appearance of color , and “Image Appearance Models” ( IAM ), in German: models for the appearance of images . Since the mathematical descriptions that only calculate colors and color distances do not take into account higher levels of human color perception, more advanced models are required, as a large number of other factors can have a strong influence on the overall impression. The developments of CAM and IAM arise from the question: “How does a certain color or an image appear in the general context of the near and far surroundings of an image?” Phenomena such as simultaneous contrast , adaptation to the ambient brightness and its temporal course, reduction in spatial resolution For twilight (mesopic vision) and dark vision (scotopic vision) play a major role in color perception.
A very common problem in this context is the fundamental contradiction between sharpness and the perception of sharpness . IAM are a step towards a solution to this conflict, as the processing of detail contrasts, color contrasts etc. is considered separately in these models.
is basically the first CAM. The adaptation to the white point is already taken into account (using transformation matrices such as von Kries or Bradford matrices), as well as the compression of the brightness perception. Further development then led to CIECAM97s.
is more precise and extensive in terms of viewing conditions, etc. The development was continued to CIECAM02.
generally provides more precise values for color differences and takes into account to a greater extent image brightness, color background, image environment, white point, adaptation and simultaneous contrast .
is another step in development. The newest representative of these models is iCAM06 . Local color adaptation, local brightness and ambient brightness, HDR , and the temporal course of the adaptation to the ambient brightness are taken into account. The area of the IAM is entered. In contrast to its predecessors, iCAM06 is already a fully-fledged IAM, since, for example, white point adaptation and contrast calculations are no longer calculated using a purely local model (pixel by pixel), but rather spatially. Thus, depending on the structure and composition, image areas can also influence more distant areas and thus change the overall impression of an image.
- Color systems in art and science (Virtual Color Museum)
- Poynton's Color FAQ (English)
- to YCC and other models
- CIE website
- Color tables and development of CAM and IAM (PDF, en: 2.85 MB)
- Meet iCAM: A Next-Generation Color Appearance Model (PDF, en: 343 kB)
- The CIE 1997 Interim Color Appearance Model (Simple Version), CIECAM97s (PDF, en: 63 kB)
- Color Appearance Models: CIECAM02 and Beyond Outline (PDF, en: 2.33 MB)
- hs-aalen.de: What is color actually? Part 1: Colorimetry , pp. 44–49, here p. 46.
- color ball according to Runge
- William David Wright, Guild
- RWG Hunt: Measuring Color, chap. 3.6 - Uniform Chromaticity Diagrams
- MacAdam: Color Measurement 2nd ed. Springer-Verlag, chap. 8.3 - Nonlinear Transformations of the Chromaticity Diagram
- http://www.brucelindbloom.com/ Navigation: Math
- CSS at W3C