Editing Luminance (section)

== Formulation ==
[[Image:Etendue.svg|right|thumb|Parameters for defining the luminance]]

The luminance of a specified point of a light source, in a specified direction, is defined by the [[mixed partial derivative]]
<math display="block">L_\mathrm{v} = \frac{\mathrm{d}^2\Phi_\mathrm{v}}{\mathrm{d}\Sigma\,\mathrm{d}\Omega_\Sigma \cos \theta_\Sigma}</math>
where
* {{math|''L''<sub>v</sub>}} is the luminance ([[candela|cd]]/[[square metre|m<sup>2</sup>]]);
* {{math|d<sup>2</sup>Φ<sub>v</sub>}} is the [[luminous flux]] ([[Lumen (unit)|lm]]) leaving the area {{math|dΣ}} in any direction contained inside the solid angle {{math|dΩ<sub>Σ</sub>}};
* {{math|dΣ}} is an [[infinitesimal]] area (m<sup>2</sup>) of the source containing the specified point;
* {{math|dΩ<sub>Σ</sub>}} is an infinitesimal [[solid angle]] ([[Steradian|sr]]) containing the specified direction; and
* {{math|''θ''<sub>Σ</sub>}} is the [[angle]] between the [[Normal (geometry)|normal]] {{math|'''n'''<sub>Σ</sub>}} to the surface {{math|dΣ}} and the specified direction.<ref>{{cite book | last = Chaves | first = Julio | title = Introduction to Nonimaging Optics, Second Edition | url = https://books.google.com/books?id=e11ECgAAQBAJ | publisher = [[CRC Press]] | year = 2015 | page = 679 | isbn = 978-1482206739 | url-status = live | archive-url = https://web.archive.org/web/20160218223513/https://books.google.com/books?id=e11ECgAAQBAJ | archive-date = 2016-02-18 }}</ref>

If light travels through a lossless medium, the luminance does not change along a given [[light ray]]. As the ray crosses an arbitrary surface {{mvar|S}}, the luminance is given by
<math display="block">L_\mathrm{v} = \frac{\mathrm{d}^2\Phi_\mathrm{v}}{\mathrm{d}S\,\mathrm{d}\Omega_S \cos \theta_S}</math>
where
* {{math|d''S''}} is the infinitesimal area of {{mvar|S}} seen from the source inside the solid angle {{math|dΩ<sub>Σ</sub>}};
* {{math|dΩ<sub>''S''</sub>}} is the infinitesimal solid angle [[Subtended angle|subtended]] by {{math|dΣ}} as seen from {{math|d''S''}}; and
* {{mvar|θ<sub>S</sub>}} is the angle between the normal {{math|'''n'''<sub>''S''</sub>}} to {{math|d''S''}} and the direction of the light.

More generally, the luminance along a light ray can be defined as
<math display="block">L_\mathrm{v} = n^2\frac{\mathrm{d}\Phi_\mathrm{v}}{\mathrm{d}G}</math>
where
* {{math|d''G''}} is the [[etendue]] of an infinitesimally narrow beam containing the specified ray;
* {{math|dΦ<sub>v</sub>}} is the luminous flux carried by this beam; and
* {{mvar|n}} is the [[index of refraction]] of the medium.