Editing Rendering equation (section)

== Equation form ==
The rendering equation may be written in the form

:<math>L_{\text{o}}(\mathbf x, \omega_{\text{o}}, \lambda, t) = L_{\text{e}}(\mathbf x, \omega_{\text{o}}, \lambda, t) + L_{\text{r}}(\mathbf x, \omega_{\text{o}}, \lambda, t)</math>
:<math>L_{\text{r}}(\mathbf x, \omega_{\text{o}}, \lambda, t) = \int_\Omega f_{\text{r}}(\mathbf x, \omega_{\text{i}}, \omega_{\text{o}}, \lambda, t) L_{\text{i}}(\mathbf x, \omega_{\text{i}}, \lambda, t) (\omega_{\text{i}}\cdot\mathbf n) \operatorname d \omega_{\text{i}}</math>

where

*<math>L_{\text{o}}(\mathbf x, \omega_{\text{o}}, \lambda, t)</math> is the total [[spectral radiance]] of wavelength <math>\lambda</math> directed outward along direction <math>\omega_{\text{o}}</math> at time <math>t</math>, from a particular position <math>\mathbf x</math>
*<math>\mathbf x</math> is the location in space
*<math>\omega_{\text{o}}</math> is the direction of the outgoing light
*<math>\lambda</math> is a particular wavelength of light
*<math>t</math> is time
*<math>L_{\text{e}}(\mathbf x, \omega_{\text{o}}, \lambda, t)</math> is [[emissivity|emitted]] spectral radiance
*<math>L_{\text{r}}(\mathbf x, \omega_{\text{o}}, \lambda, t)</math> is [[reflectance|reflected]] spectral radiance
*<math>\int_\Omega \dots \operatorname d\omega_{\text{i}}</math> is an [[integral]] over <math>\Omega</math>
*<math>\Omega</math> is the unit [[Sphere|hemisphere]] centered around <math>\mathbf n</math> containing all possible values for <math>\omega_{\text{i}}</math> where <math>\omega_{\text{i}}\cdot\mathbf n > 0</math>
*<math>f_{\text{r}}(\mathbf x, \omega_{\text{i}}, \omega_{\text{o}}, \lambda, t)</math> is the [[bidirectional reflectance distribution function]], the proportion of light reflected from <math>\omega_{\text{i}}</math> to <math>\omega_{\text{o}}</math> at position <math>\mathbf x</math>, time <math>t</math>, and at wavelength <math>\lambda</math>
*<math>\omega_{\text{i}}</math> is the negative direction of the incoming light
*<math>L_{\text{i}}(\mathbf x, \omega_{\text{i}}, \lambda, t)</math> is spectral radiance of wavelength <math>\lambda</math> coming inward toward <math>\mathbf x</math> from direction <math>\omega_{\text{i}}</math> at time <math>t</math>
*<math>\mathbf n</math> is the [[Normal (geometry)|surface normal]] at <math>\mathbf x</math>
*<math>\omega_{\text{i}} \cdot \mathbf n</math> is the weakening factor of outward [[irradiance]] due to [[angle of incidence (optics)|incident angle]], as the light flux is smeared across a surface whose area is larger than the projected area perpendicular to the ray. This is often written as <math>\cos \theta_i</math>.

Two noteworthy features are: its linearity—it is composed only of multiplications and additions, and its spatial homogeneity—it is the same in all positions and orientations. These mean a wide range of factorings and rearrangements of the equation are possible. It is a [[Fredholm integral equation]] of the second kind, similar to those that arise in [[quantum field theory]].<ref>{{cite book| last1=Watt| first1=Alan| last2=Watt| first2=Mark |title=Advanced Animation and Rendering Techniques: Theory and Practice| url=https://archive.org/details/advancedanimatio00watt| url-access=limited|year=1992|publisher=Addison-Wesley Professional|isbn=978-0-201-54412-1|page=[https://archive.org/details/advancedanimatio00watt/page/n299 293]|section=12.2.1 The path tracing solution to the rendering equation}}</ref>

Note this equation's [[spectrum|spectral]] and [[time]] dependence — <math>L_{\text{o}}</math> may be sampled at or integrated over sections of the [[visible spectrum]] to obtain, for example, a [[trichromatic]] color sample. A pixel value for a single frame in an animation may be obtained by fixing <math>t;</math> [[motion blur]] can be produced by [[averaging]] <math>L_{\text{o}}</math> over some given time interval (by integrating over the time interval and dividing by the length of the interval).<ref>{{cite web | last = Owen | first = Scott | title = Reflection: Theory and Mathematical Formulation | date = September 5, 1999 | url = http://www.siggraph.org/education/materials/HyperGraph/illumin/reflect2.htm | accessdate = 2008-06-22}}</ref>

Note that a solution to the rendering equation is the function <math>L_{\text{o}}</math>. The function <math>L_{\text{i}}</math> is related to <math>L_{\text{o}}</math> via a ray-tracing operation: The incoming radiance from some direction at one point is the outgoing radiance at some other point in the opposite direction.