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| [[File:PlanckianLocus.png|right|300px|thumb|Planckian locus in the CIE 1931 chromaticity diagram]]
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| In [[physics]] and [[color science]], the '''Planckian locus''' or '''black body locus''' is the path or [[locus (mathematics)|''locus'']] that the color of an [[incandescent]] [[black body]] would take in a particular [[chromaticity space]] as the blackbody [[temperature]] changes. It goes from deep [[red]] at low temperatures through [[orange (color)|orange]], [[yellow]]ish white, [[white]], and finally [[blue|bluish]] white at very high temperatures.
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| A [[color space]] is a [[three-dimensional space]]; that is, a color is specified by a set of three numbers (the [[CIE 1931 color space|CIE]] coordinates ''X'', ''Y'', and ''Z'', for example, or other values such as [[hue]], [[colorfulness]], and [[luminance]]) which specify the color and brightness of a particular homogeneous visual stimulus. A chromaticity is a color projected into a [[two-dimensional space]] that ignores brightness. For example, the standard [[CIE XYZ color space]] projects directly to the corresponding chromaticity space specified by the two chromaticity coordinates known as ''x'' and ''y'', making the familiar chromaticity diagram shown in the figure. The Planckian locus, the path that the color of a black body takes as the blackbody temperature changes, is often shown in this standard chromaticity space.
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| == The Planckian locus in the XYZ color space ==
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| [[File:CIE 1931 XYZ Color Matching Functions.svg|thumb|right|[http://www.cie.co.at/main/freepubs.html CIE 1931 Standard Colorimetric Observer] functions used to map blackbody spectra to XYZ coordinates]]
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| In the [[CIE 1931 color space|CIE XYZ color space]], the three coordinates defining a color are given by ''X'', ''Y'', and ''Z'':<ref name=stiles>{{cite book | author=Wyszecki, Günter and Stiles, Walter Stanley | title=Color Science: Concepts and Methods, Quantitative Data and Formulae |edition=2E| publisher=Wiley-Interscience | year=2000 | isbn=0-471-39918-3}}</ref>
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| :<math>X_T = \int_0^\infty X(\lambda)I(\lambda,T)\,d\lambda</math> | |
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| :<math>Y_T = \int_0^\infty Y(\lambda)I(\lambda,T)\,d\lambda</math>
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| :<math>Z_T = \int_0^\infty Z(\lambda)I(\lambda,T)\,d\lambda</math>
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| where ''I''(λ,T) is the spectral [[radiance]] of the light being viewed, and ''X''(''λ''), ''Y''(''λ'') and ''Z''(''λ'') are the [[color matching function]]s 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]]:
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| :<math>I(\lambda,T) =\frac{2 hc^2}{\lambda^5}\frac{1}{\exp\left(\frac{hc/\lambda}{kT}\right)-1}</math>
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| where:
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| :''I'' is the black body spectral radiance (power per unit area per unit solid angle per unit wavelength)
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| :''T'' is the [[temperature]] of the black body
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| :''h'' is [[Planck's constant]]
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| :''c'' is the [[speed of light]]
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| :''k'' is [[Boltzmann's constant]]
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| This will give the Planckian locus in CIE XYZ color space. If these coordinates are ''X<sub>T</sub>'', ''Y<sub>T</sub>'', ''Z<sub>T</sub>'' where ''T'' is the temperature, then in the CIE chromaticity coordinates will be
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| :<math>x_T = \frac{X_T}{X_T+Y_T+Z_T}</math>
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| :<math>y_T = \frac{Y_T}{X_T+Y_T+Z_T}</math>
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| ===Approximation===
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| The Planckian locus in ''xy'' space is depicted as a curve in the chromaticity diagram above. While it is possible to compute the CIE ''xy'' co-ordinates exactly given the above formulas, it is faster to use approximations. Since the [[mired]] scale changes more evenly along the locus than the temperature itself, it is common for such approximations to be functions of the reciprocal temperature. Kim ''et al.'' uses a [[cubic spline]]:<ref>{{US patent reference | |
| | number = 7024034
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| | y = 2006
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| | m = 04
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| | d = 04
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| | inventor = Kim ''et al.''
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| | title = Color Temperature Conversion System and Method Using the Same
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| }}</ref><ref>{{cite journal|journal=Journal of the Korean Physical Society|volume=41|issue=6|date=December 2002|pages=865–871|title=Design of Advanced Color Temperature Control System for HDTV Applications| url=http://icpr.snu.ac.kr/resource/wop.pdf/J01/2002/041/R06/J012002041R060865.pdf|author=Bongsoon Kang, Ohak Moon, Changhee Hong, Honam Lee, Bonghwan Cho and Youngsun Kim}}</ref>
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| <math>x_c=\begin{cases}
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| -0.2661239 \frac{10^9}{T^3} - 0.2343580 \frac{10^6}{T^2} + 0.8776956 \frac{10^3}{T} + 0.179910 & 1667\text{K} \leq T \leq 4000\text{K} \\
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| -3.0258469 \frac{10^9}{T^3}+2.1070379 \frac{10^6}{T^2} + 0.2226347 \frac{10^3}{T} + 0.240390 & 4000\text{K} \leq T \leq 25000\text{K}
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| \end{cases}</math>
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| <math>y_c=\begin{cases} | |
| -1.1063814 x_c^3 - 1.34811020 x_c^2 + 2.18555832 x_c - 0.20219683 & 1667\text{K} \leq T \leq 2222\text{K} \\
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| -0.9549476 x_c^3 - 1.37418593 x_c^2 + 2.09137015 x_c - 0.16748867 & 2222\text{K} \leq T \leq 4000\text{K} \\
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| +3.0817580 x_c^3 - 5.87338670 x_c^2 + 3.75112997 x_c - 0.37001483 & 4000\text{K} \leq T \leq 25000\text{K}
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| \end{cases}</math>
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| [[File:Planckian-locus-approximation.png|thumb|Kim ''et al.'''s approximation to the Planckian locus (shown in red). The notches demarcate the three splines (shown in blue).]]
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| The inverse calculation, from chromaticity co-ordinates (''x'',''y'') on or near the Planckian locus to correlated color temperature, is discussed in [[Color temperature#Approximation]].
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| The Planckian locus can also be approximated in the CIE 1960 UCS, which is used to compute CCT and CRI, using the following expressions:<ref>{{cite journal| journal=Color Research & Application|title=An algorithm to calculate correlated colour temperature|first=Michael P.|last=Krystek|volume=10|issue=1|date=January 1985|pages=38–40|doi=10.1002/col.5080100109|quote=A new algorithm to calculate correlated colour temperature is given. This algorithm is based on a rational Chebyshev approximation of the Planckian locus in the CIE 1960 UCS diagram and a bisection procedure. Thus time-consuming search procedures in tables or charts are no longer necessary.}}</ref>
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| <math>\bar{u}(T)=\frac{0.860117757+1.54118254 \times 10^{-4}T + 1.28641212 \times 10^{-7} T^2}{1+8.42420235 \times 10^{-4}T + 7.08145163 \times 10^{-7}T^2}</math>
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| <math>\bar{v}(T)=\frac{0.317398726+4.22806245 \times 10^{-5}T + 4.20481691 \times 10^{-8} T^2}{1-2.89741816 \times 10^{-5}T+1.61456053 \times 10^{-7}T^2}</math>
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| This approximation is accurate to within <math>\left| u-\bar{u} \right| < 8\times10^{-5}</math> and <math>\left|v-\bar{v}\right|<9\times10^{-5}</math> for <math>1000K<T<15,000K</math>
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| ==Correlated color temperature==
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| {{quote|The '''correlated color temperature''' (T<sub>cp</sub>) is the temperature
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| of the Planckian radiator whose perceived colour most closely resembles that of a given stimulus at the same brightness and under specified viewing conditions| [http://www.cie.co.at/publ/abst/17-4-89.html CIE/IEC 17.4:1987]|International Lighting Vocabulary (ISBN 3900734070)<ref>{{cite journal|title=The concept of correlated colour temperature revisited|first=Ákos|last=Borbély|coauthors=Sámson,Árpád;Schanda, János|volume=26|issue=6|pages=450–457|date=December 2001|doi=10.1002/col.1065|journal=Color Research & Application| url=http://www.knt.vein.hu/staff/schandaj/SJCV-Publ-2005/462.doc}}</ref>}} | |
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| 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 the [[CIE 1960 color space]], also known as MacAdam's (u,v) diagram.<ref>{{cite journal|title=Lines of constant correlated color temperature based on MacAdam's (u,v) Uniform chromaticity transformation of the CIE diagram|first=Kenneth L.|last=Kelly|journal=[[JOSA]]|volume=53|issue=8|date=August 1963| url=http://www.opticsinfobase.org/abstract.cfm?URI=josa-53-8-999|format=abstract|doi=10.1364/JOSA.53.000999|pages=999}}</ref> Today, the CIE 1960 color space is deprecated for other purposes:<ref>{{cite book|title=Lighting Engineering: Applied Calculations|first=Ronald Harvey|last=Simons|coauthors=Bean, Arthur Robert|publisher=Architectural Press|isbn=0-7506-5051-6|year=2001| url=http://books.google.com/?id=SWzBKDGHxOUC&pg=PA289&dq=%22correlated+colour+temperature%22+CIE+macadam}}</ref>
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| {{quote|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)|[http://www.cie.co.at/publ/abst/13-3-95.html Method of Measuring and Specifying Colour Rendering Properties of Light Sources]}}
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| Owing to the perceptual inaccuracy inherent to the concept, it suffices to calculate to within 2K at lower CCTs and 10K at higher CCTs to reach the threshold of imperceptibility.<ref>{{cite web|title=Results of the Intercomparison of Correlated Color Temperature Calculation| url=http://cie2.nist.gov/CR3/Documents/Results_CCTcomparison.pdf|date=19 June 1999|publisher=CORM|first=Yoshi|last=Ohno|coauthors=Jergens, Michael}}</ref>
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| [[File:Planckian-locus.png|600px|thumb|center|Close up of the [[CIE 1960 color space|CIE 1960 UCS]]. The isotherms are perpendicular to the Planckian locus, and are drawn to indicate the maximum distance from the locus that the CIE considers the correlated color temperature to be meaningful: <math>\Delta_{uv}=\pm 0.05</math>]]
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| ===International Temperature Scale===
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| The Planckian locus is derived by the determining the chromaticity values of a Planckian radiator using the standard colorimetric observer. The relative [[spectral power distribution]] (SPD) of a Planckian radiator follows Planck's law, and depends on the second radiation constant, <math>c_2=hc/k</math>. 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 illuminant]]s, 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:<ref>{{cite book|title=Colorimetry: Understanding the CIE System|author=Janos Schanda|publisher=[[Wiley Interscience]]|year=2007|chapter=3: CIE Colorimetry|isbn=978-0-470-04904-4|pages=37–46}}</ref><ref>[http://www.its-90.com/its-90p4.html The ITS-90 Resource Site]</ref>
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| * <math>c_2=1.432 \times 10^{-2} \text{m·K}</math> (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<sup>-2</sup><ref>{{cite journal|title=The Early History of the International Practical Scale of Temperature|first=J.A.|last=Hall|doi=10.1088/0026-1394/3/1/006|journal=Metrologia|volume=3|issue=1|pages=25–28|date=January 1967}}</ref><ref>{{cite journal|title=A table of Planckian radiation|first=Parry|last=Moon|date=March 1948|journal=[[JOSA]]|volume=38|issue=3|pages=291–294|url=http://www.opticsinfobase.org/abstract.cfm?URI=josa-38-3-291|format=abstract|doi=10.1364/JOSA.38.000291}}</ref>
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| * <math>c_2=1.4380 \times 10^{-2} \text{m·K}</math> (IPTS-48). In effect for Illuminant series D (formalized in 1967).
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| * <math>c_2=1.4388 \times 10^{-2} \text{m·K}</math> (ITS-68), (ITS-90). Often used in recent papers.
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| * <math>c_2=1.4387752(25) \times 10^{-2} \text{m·K}</math> ([[CODATA]], 2006). Current value, as of 2008.<ref>{{cite web|title=CODATA Recommended Values of the Fundamental Physical Constants: 2006|year=2007|first=Peter J.|last=Mohr|coauthors=Taylor, Barry N. and Newell, David B.|url=http://physics.nist.gov/cuu/Constants/codata.pdf}}</ref>
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| ==References==
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| <references/>
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| ==External links==
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| *[http://www.vendian.org/mncharity/dir3/blackbody/UnstableURLs/bbr_color.html Numerical table of color temperature and the corresponding xy and sRGB coordinates for both the 1931 and 1964 CMFs], by Mitchell Charity.
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| [[Category:Color space]]
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