Planckian_locus

In color theory, the Planckian locus is generally the path or locus that the color of a black body would take in a particular color space as the blackbody temperature changes. Generally, a color space is a set of three numbers (X, Y, and Z, for example) which specify the color and brightness of a particular homogeneous visual stimulus. Sometimes we may only wish to deal with the chromaticity (color) of a visual stimulus. This is a two-dimensional space of two numbers (x and y) which leave out the brightness information. In this case the Planckian locus is the path that the color of a black body takes in this chromaticity space as the blackbody temperature changes.

The Planckian locus in the XYZ color space

CIE 1964 Supplementary Standard Colorimetric Observer functions X, Y, and Z functions between 380 nm and 780 nm (at 5 nm intervals).

In the CIE XYZ color space, the three coordinates defining a color are given by X, Y, and Z where

$X\; =\; \backslash int\_0^\backslash infty\; X(\backslash lambda)I(\backslash lambda)\backslash ,d\backslash lambda$

$Y\; =\; \backslash int\_0^\backslash infty\; Y(\backslash lambda)I(\backslash lambda)\backslash ,d\backslash lambda$

$Z\; =\; \backslash int\_0^\backslash infty\; Z(\backslash lambda)I(\backslash lambda)\backslash ,d\backslash lambda$

where I(λ) is the spectral radiance of the light being viewed, and X(λ), Y(λ) and Z(λ) are the color matching functions of the CIE standard colorimetric observer, shown in the diagram on the right, and λ is the wavelength. The Planckian locus is determined by substituting into the above equations the black body spectral radiance, which is given by Planck\'s law:

$I(\backslash lambda)\; =\backslash frac\{2\backslash pi\; hc^2\}\{\backslash lambda^5\}\backslash frac\{1\}\{\backslash exp\backslash left(\backslash frac\{hc/\backslash lambda\}\{kT\}\backslash right)-1\}$

where:

I is the black body spectral radiance (power per unit area per unit solid angle per unit wavelength)

T is the temperature of the black body

h is Planck\'s constant

c is the speed of light

k is Boltzmann\'s constant

Planckian locus in the CIE 1931 chromaticity diagram

This will give the Planckian locus in CIE XYZ color space. If these coordinates are X(T), Y(T), Z(T) where T is the temperature, then in the CIE chromaticity coordinates will be

$x(T)\; =\; \backslash frac\{X(T)\}\{X(T)+Y(T)+Z(T)\}$

$y(T)\; =\; \backslash frac\{Y(T)\}\{X(T)+Y(T)+Z(T)\}$

The Planckian locus in xy space is shown in the chromaticity diagram on the right. While it is possible to compute the CIE xy(Y) coordinates exactly given the above formulas, it is faster to compute these using polynomial approximations.

Correlated color temperature

The correlated color temperature (T_cp) is the temperature of the Planckian radiator whose perceived colour most closely resembles that of a given stimulus at the same brightness and under specified viewing conditions

— CIE/IEC 17.4:1987, International Lighting Vocabulary (ISBN 3900734070)Borbély, Ákos; Sámson,Árpád;Schanda, János (December 2001). "The concept of correlated colour temperature revisited". Color Research & Application 26 (6): 450-457. doi:10.1002/col.1065. 

The mathematical procedure for determining the correlated color temperature involves finding the closest point to the light source\'s white point on the Planckian locus. Since the CIE\'s 1959 meeting in Brussels, the Planckian locus has been computed using MacAdam\'s 1937 (u,v) diagram (uniform color space).Kelly, Kenneth L. (August 1963). "Lines of constant correlated color temperature based on MacAdam\'s (u,v) Uniform chromaticity transformation of the CIE diagram". JOSA 53 (8).  Today, the CIE 1960 color space is deprecated for other purposes:Simons, Ronald Harvey; Bean, Arthur Robert (2001). Lighting Engineering: Applied Calculations. Architectural Press. ISBN 0750650516. 

The 1960 UCS diagram and 1964 Uniform Space are declared obsolete recommendation in CIE 15.2 (1986), but have been retained for the time being for calculating colour rendering indices and correlated colour temperature.

—CIE 13.3 (1995), Method of Measuring and Specifying Colour Rendering Properties of Light Sources

Owing to the perceptual inaccuracy inherent to the concept, the Council for Optical Radiation Measurement has concluded that it suffices to calculate to within 2K at lower CCTs and 10K at higher CCTs to reach the imperceptibility threshold.Ohno, Yoshi; Jergens, Michael (June 19 1999). Results of the Intercomparison of Correlated Color Temperature Calculation. CORM.

International Temperature Scale

The Planckian locus is derived by the determining the chromaticity values of a Planckian radiator using the standard colorimetric observer. The relative SPD of Planckian radiator follows Planck\'s law, and depends on the second radiation constant, $c\_2=hc/k$. As measuring techniques have improved, the General Conference on Weights and Measures has revised its estimate of this constant, with the International Temperature Scale (and briefly, the International Practical Temperature Scale). These successive revisions caused a shift in the Planckian locus and, as a result, the correlated color temperature scale. Before ceasing publication of standard illuminants, the CIE worked around this problem by explicitly specifying the form of the SPD, rather than making references to black bodies and a color temperature. Nevertheless, it is useful to be aware of previous revisions in order to be able to verify calculations made in older texts:Janos Schanda (2007). "3: CIE Colorimetry", Colorimetry: Understanding the CIE System. Wiley Interscience, 37-46. ISBN 978-0-470-04904-4. The ITS-90 Resource Site

• $c\_2=1.432\; \backslash times\; 10^\{-2\}\; \backslash text\{m·K\}$ (ITS-27). Note: Was in effect during the standardization of Illuminants A, B, C (1931), however the CIE used the value recommended by the U.S. National Bureau of Standards, 1.435 × 10^-2Hall, J.A. (January 1967). "The Early History of the International Practical Scale of Temperature". Metrologia 3: 25-28. doi:10.1088/0026-1394/3/1/006. Moon, Parry. "A table of Planckian radiation". JOSA 38 (3): 291-294. 
• $c\_2=1.4380\; \backslash times\; 10^\{-2\}\; \backslash text\{m·K\}$ (IPTS-48). In effect for Illuminant series D (formalized in 1967).
• $c\_2=1.4388\; \backslash times\; 10^\{-2\}\; \backslash text\{m·K\}$ (ITS-68), (ITS-90). Often used in recent papers.
• $c\_2=1.4387752(25)\; \backslash times\; 10^\{-2\}\; \backslash text\{m·K\}$ (CODATA, 2006). Current value, as of 2008.Mohr, Peter J.; Taylor, Barry N. and Newell, David B. (2007). CODATA Recommended Values of the Fundamental Physical Constants: 2006.

References

• Wyszecki, Günter and Stiles, W.S. (2000). Color Science - Concepts and Methods, Quantitative Data and Formulae, 2nd edition, Wiley-Interscience. ISBN 0-471-39918-3. 

External links

• Numerical table of color temperature and the corresponding xy and sRGB coordinates for both the 1931 and 1964 CMFs, by Mitchell Charity.
• Planckian xy locus for 1000K-25000K using the 2° CMF, in 1K increments, by Timo Autiokari.

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