# Encyclopedia of Color Science and Technology

2016 Edition
| Editors: Ming Ronnier Luo

# CIE Chromaticity Diagrams, CIE Purity, CIE Dominant Wavelength

Reference work entry
DOI: https://doi.org/10.1007/978-1-4419-8071-7_325

## Definitions

### Chromaticity Diagram

Plane diagram in which points specified by chromaticity coordinates represent the chromaticities of color stimuli [1]

Note: In the CIE standard colorimetric systems, y is normally plotted as ordinate and x as abscissa, to obtain an x, y chromaticity diagram.

### Purity (of a Color Stimulus)

Measure of the proportions of the amounts of the monochromatic stimulus and of the specified achromatic stimulus that, when additively mixed, match the color stimulus considered.

Note 1: In the case of purple stimuli, the monochromatic stimulus is replaced by a stimulus whose chromaticity is represented by a point on the purple boundary.

Note 2: The proportions can be measured in various ways (see “Excitation Purity” and “Colorimetric Purity”).

### Excitation Purity [pe]

Quantity defined by the ratio NC/ND of two collinear distances on the chromaticity diagram of the CIE 1931 or 1964 standard colorimetric systems, the first distance being that between the point C representing the color stimulus considered and the point N representing the specified achromatic stimulus and the second distance being that between the point N and the point D on the spectrum locus at the dominant wavelength of the color stimulus considered, leading to the following expressions:
$${p}_{\mathrm{e}}=\frac{y-{y}_{\mathrm{n}}}{y_{\mathrm{d}}-{y}_{\mathrm{n}}}$$
or
$${p}_{\mathrm{e}}=\frac{x-{x}_{\mathrm{n}}}{x_{\mathrm{d}}-{x}_{\mathrm{n}}}$$
where (x, y), (xn, yn), (xd, yd,) are the x, y chromaticity coordinates of the points C, N, and D, respectively.

Unit: 1

Note 1: In the case of purple stimuli, see Note 1 to “purity.”

Note 2: The formulae in x and y are equivalent, but greater precision is given by the formula which has the greater value in the numerator.

Note 3: Excitation purity, pe, is related to colorimetric purity, pc, by the equation:
$${p}_{\mathrm{e}}=\frac{p_{\mathrm{c}}y}{y_{\mathrm{d}}}$$

### Colorimetric Purity [pc]

Quantity defined by the relation
$${p}_{\mathrm{c}}=\frac{L_{\mathrm{d}}}{L_{\mathrm{n}}+{L}_{\mathrm{d}}}$$
where Ld and Ln are the respective luminances of the monochromatic stimulus and of the specified achromatic stimulus that match the color stimulus considered in an additive mixture.

Note 1: In the case of purple stimuli, see Note 1 to “purity.”

Note 2: In the CIE 1931 standard colorimetric system, colorimetric purity, pc, is related to excitation purity, pe, by the equation $${p}_{\mathrm{c}}={p}_{\mathrm{e}}\;{y}_{\mathrm{d}}/y$$ where yd and y are the y chromaticity coordinates, respectively, of the monochromatic stimulus and the color stimulus considered.

Note 3: In the CIE 1964 standard colorimetric system, a measure, pc,10, is defined by the relation given in Note 2, but using pe,10, yd,10, and y10 instead of pe, yd, and y, respectively.

### Dominant Wavelength (of a Color Stimulus) [λd]

Wavelength of the monochromatic stimulus that, when additively mixed in suitable proportions with the specified achromatic stimulus, matches the color stimulus considered in the CIE 1931 x, y chromaticity diagram.

Unit: nm

Note: In the case of purple stimuli, the dominant wavelength is replaced by the complementary wavelength.

## Overview

### x, y Chromaticity Diagram

Both in the CIE 1931 standard colorimetric system and the CIE 1964 standard colorimetric system, chromaticity coordinates are expressed as the ratio of the given tristimulus value and the sum of all three tristimulus values [2]:
$$\begin{array}{ll}x=\frac{X}{X+Y+Z},\hfill & y=\frac{Y}{X+Y+Z}\hfill \end{array}.$$
(1)
As the color-matching functions are the tristimulus values of the monochromatic stimuli, the chromaticity coordinates of the monochromatic stimuli can be calculated according to Eq. 1 In the plane rectangular x-y diagram the line of the chromaticity of the monochromatic stimuli bounds, together with the straight line connecting the red and blue endpoints of the spectrum, the area of visible stimuli [3]; see Fig. 1. The diagram produced by plotting x as abscissa and y as ordinate is called the CIE 1931 chromaticity diagram or the CIE (x, y) diagram. A similar chromaticity diagram can be constructed using the x10, y10 chromaticity coordinates of the CIE 1964 standard colorimetric system.

The chromaticity diagram is often depicted in color; see Fig. 3 (in the section for dominant wavelength and purity). One has to emphasize, however, that in this figure, the colors are only for orientation. As shown in Fig. 1, if, e.g., on the computer the colors are mixed from the R, G, B primaries (the gamut of real RGB primaries of monitors is even smaller), the mixed colors have to be inside the RGB triangle. On the boundary of the monochromatic stimuli, the emission spectrum reaching our eyes should contain only one single wavelength [4].

As seen in Fig. 3, the mid-part of the chromaticity diagram looks whitish. This is even more pronounced if a light source of that chromaticity illuminates a scene; a white paper will – under these conditions – look white, and this is caused by chromatic adaptation.

Chromaticity diagrams can be built also for the CIE 1976 u′, v′ coordinates [5]. The 1976 u′, v′ uniform chromaticity scale diagram (UCS diagram) is a projective transformation of the CIE 1931 x, y chromaticity diagram yielding perceptually more uniform color spacing, i.e., the perceived chromaticity differences are represented by more uniform coordinate differences. The transformation between the two systems is
$$\begin{array}{l}{u}^{\prime }=\frac{4x}{-2x+12y+3}\hfill \\ {}{v}^{\prime }=\frac{9y}{-2x+12y+3}\hfill \end{array}.$$
(2)
With these coordinates, the chromaticity diagram has the form as shown in Fig. 2. Comparing this diagram with the x, y diagram, it becomes obvious how nonuniform the x, y diagram is (see details in entry “”).
An equivalent transformation starting from the tristimulus values is
$$\begin{array}{l}{u}^{\prime }=\frac{4X}{X+15Y+3Z}\hfill \\ {}{v}^{\prime }=\frac{9Y}{X+15Y+3Z}\hfill \end{array}.$$
(3)

To be exact Euclidean distances in his diagram can be used to represent approximately the relative perceived magnitude of color differences between color stimuli of negligibly different luminances, of approximately the same size, and viewed in identical surroundings, by an observer photopically adapted to a field with the chromaticity of CIE standard illuminant D65 [6].

## Dominant Wavelength and Purity

A color can be characterized by its tristimulus values or its chromaticity and the luminance (if it is a self-luminous object) or luminance factor (if it is a reflecting or transmitting object illuminated by a (standard) light source). It is difficult to visualize the chromaticity from the x, y values; an easier identification is by two other quantities: dominant wavelength and excitation purity.

### Dominant and Complementary Wavelength

In Fig. 3, we see two colored samples (represented in the chromaticity diagram by A and B); they are illuminated by a source of neutral chromaticity (N). If a line is drawn from point N through point A or B, one reaches at the boundaries of the chromaticity diagram, at the spectrum locus points D and C, respectively. Points D and A are located on the same side of point N; thus, chromaticity of A is less saturated as that of D but has similar hue; therefore, the wavelength of the monochromatic radiation at point D is called the dominant wavelength (in our example 495 nm). By mixing radiation of the monochromatic radiation D and the neutral radiation N, one can create the chromaticity A.

For point B, as it is located on the far side of points N and C, one can produce the chromaticity N by mixing chromaticity B with C. Therefore, the wavelength of the spectral line at C is called the complementary wavelength for chromaticity B.

## Excitation Purity

The relative distance of A (resp. B) from N, compared to the distances $$\overline{\mathrm{DN}}$$ (resp. $$\overline{{\mathrm{B}}^{\prime}\mathrm{N}}$$), is called excitation purity and describes how strongly the monochromatic stimulus is diluted by the radiation of N. For purple colors (in the triangle of points N-V-R), the monochromatic stimulus is replaced, as seen, by the stimulus on the purple boundary. In practice it is not necessary to calculate with the vector length, it is enough to take either the x or the y coordinates. One should take always those coordinates that are larger; thus, e.g., for the two chromaticity points A and B, the excitation purities are calculated as
$$\mathrm{Excitation}\ \mathrm{purity}\ \mathrm{of}\ \mathrm{chromaticity}\ \mathrm{point}\ \mathrm{A}:{p}_{\mathrm{e},\mathrm{A}}=\frac{x_{\mathrm{A}}-{x}_{\mathrm{N}}}{x_{\mathrm{D}}-{x}_{\mathrm{N}}}$$
(4)
$$\mathrm{Excitation}\ \mathrm{purity}\ \mathrm{of}\ \mathrm{chromaticity}\ \mathrm{point}\ \mathrm{B}:{p}_{\mathrm{e},\mathrm{B}}=\frac{y_{\mathrm{B}}-{y}_{\mathrm{N}}}{y_{{\mathrm{B}}^{\prime }}-{y}_{\mathrm{N}}}$$
(5)

## Colorimetric Purity

As mentioned under Definitions, another purity quantity, colorimetric purity, is defined by the luminance of the respective stimuli: Given the stimulus A, to mix this color from stimuli D and N, one needs luminance LD and LN. With these quantities, the colorimetric purity is
$${p}_c=\frac{L_{\mathrm{D}}}{L_{\mathrm{N}}+{L}_{\mathrm{D}}}.$$
(6)

## Summary

With some practice, one gets a reasonable feeling of the chromaticity of monochromatic stimuli if their wavelength is given; thus, if the dominant/complementary wavelength of a stimulus is stated, one can form a mental picture of the stimulus. Similarly the excitation purity is also a relatively easily visualized quantity – how whitish the given colored stimulus is – thus, these two quantities are often used instead of the chromaticity coordinates for a quick description of the chromaticity of a stimulus. One has to emphasize – however – that the chromaticity of the neutral stimulus is important. In many colorimetric calculations, the CIE standard illuminant D65 is used as a reference neutral stimulus, but in some applications, the equienergetic stimulus is found.

The CIE 1931 x, y chromaticity diagram is the most often used diagram. It is, however, non-equidistant, i.e., in different parts of the chromaticity diagram, perceived equal chromaticity differences are observed as different coordinate differences. The CIE 1976 u′, v′ diagram is more equidistant and is generally used in lighting engineering.

There is one exception, the determination of correlated color temperature, which is determined in the CIE 1960 diagram, the coordinates of which are the following:
$$\begin{array}{ll}u={u}^{\prime },\hfill & v=\frac{2}{3}{v}^{\prime}\hfill \end{array}.$$
(7)
For further details, see Eq. 2.

## Cross-References

### References

1. 1.
Commission Internationale d’Eclairage: International lighting vocabulary. CIE S 017/E:2011. see also http://eilv.cie.co.at
2. 2.
Commission Internationale d’Eclairage: Colorimetry – Part 3: CIE Tristimulus Values. CIE S 14-3/E (2011)Google Scholar
3. 3.
Commission Internationale d’Eclairage: Colorimetry, 3rd edn. CIE 015 (2004)Google Scholar
4. 4.
Schanda, J.: CIE colorimetry, Chap 3. In: Schanda, J. (ed.) CIE Colorimetry – Understanding the CIE System. Wiley Interscience (2007)Google Scholar
5. 5.
Commission Internationale d’Eclairage: Colorimetry – Part 5: CIE 1976 L*u*v* Colour Space and u′, v′ Uniform Chromaticity Scale Diagram. CIE S 014-5/E (2009)Google Scholar
6. 6.
Commission Internationale d’Eclairage: Colorimetry – Part 2: CIE Standard Illuminants. CIE S 124-2/E (2007)Google Scholar