The rendering equation may be written in the form :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) :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}} where • L_{\text{o}}(\mathbf x, \omega_{\text{o}}, \lambda, t) is the total
spectral radiance of wavelength \lambda directed outward along direction \omega_{\text{o}} at time t, from a particular position \mathbf x • \mathbf x is the location in space • \omega_{\text{o}} is the direction of the outgoing light • \lambda is a particular wavelength of light • t is time • L_{\text{e}}(\mathbf x, \omega_{\text{o}}, \lambda, t) is
emitted spectral radiance • L_{\text{r}}(\mathbf x, \omega_{\text{o}}, \lambda, t) is
reflected spectral radiance • \int_\Omega \dots \operatorname d\omega_{\text{i}} is an
integral over \Omega • \Omega is the unit
hemisphere centered around \mathbf n containing all possible values for \omega_{\text{i}} where \omega_{\text{i}}\cdot\mathbf n > 0 • f_{\text{r}}(\mathbf x, \omega_{\text{i}}, \omega_{\text{o}}, \lambda, t) is the
bidirectional reflectance distribution function, the proportion of light reflected from \omega_{\text{i}} to \omega_{\text{o}} at position \mathbf x, time t, and at wavelength \lambda • \omega_{\text{i}} is the negative direction of the incoming light • L_{\text{i}}(\mathbf x, \omega_{\text{i}}, \lambda, t) is spectral radiance of wavelength \lambda coming inward toward \mathbf x from direction \omega_{\text{i}} at time t • \mathbf n is the
surface normal at \mathbf x • \omega_{\text{i}} \cdot \mathbf n is the weakening factor of outward
irradiance due to
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 \cos \theta_i. 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. Note this equation's
spectral and
time dependence — L_{\text{o}} 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 t;
motion blur can be produced by
averaging L_{\text{o}} over some given time interval (by integrating over the time interval and dividing by the length of the interval). Note that a solution to the rendering equation is the function L_{\text{o}}. The function L_{\text{i}} is related to L_{\text{o}} 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. == Applications ==