Editing Rendering equation (section)

{{short description|Integral equation}}
[[File:Rendering eq.png|right|thumb|300px|The rendering equation describes the amount of light leaving a point {{mvar|x}} along a particular viewing direction, given functions for incoming light and emitted light, and a [[Bidirectional reflectance distribution function|BRDF]].]]

In [[computer graphics]], the '''rendering equation''' is an [[integral equation]] that expresses the amount of light leaving a point on a surface as the sum of emitted light and reflected light. It was independently introduced into computer graphics by David Immel et al.<ref name="Immel1986">
 {{Cite conference <!-- Citation bot no -->| last1 = Immel | first1 = David S.
 | last2 = Cohen | first2 = Michael F.
 | last3 = Greenberg | first3 = Donald P.
 | conference = Proceedings of the 13th annual conference on Computer graphics and interactive techniques |title= SIGGRAPH '86
 | chapter = A radiosity method for non-diffuse environments
 | chapter-url = http://www0.cs.ucl.ac.uk/research/vr/Projects/VLF/vlfpapers/multi-pass_hybrid/Immel_D_S__A_Radiosity_Method_for_Non-Diffuse_Environments.pdf
 | editor1= David C. Evans | editor2= RussellJ. Athay
 | doi = 10.1145/15922.15901
 | year = 1986
 | pages = 133–142
 | isbn = 978-0-89791-196-2| s2cid = 7384510
 }}</ref> and [[James Kajiya]]<ref name="Kajiya1986">
 {{Cite conference <!-- Citation bot no -->| last1 = Kajiya | first1 = James T.
 | conference = Proceedings of the 13th annual conference on Computer graphics and interactive techniques
 | chapter = The rendering equation
 | title= SIGGRAPH '86
 | chapter-url = http://www.cse.chalmers.se/edu/year/2011/course/TDA361/2007/rend_eq.pdf
 | doi = 10.1145/15922.15902
 | year = 1986
 | editor1= David C. Evans | editor2= RussellJ. Athay
 | pages = 143–150
 | isbn = 978-0-89791-196-2| s2cid = 9226468
 }}</ref> in 1986. The equation is important in the theory of [[physically based rendering]], describing the relationships between the [[Bidirectional reflectance distribution function|bidirectional reflectance distribution function (BRDF)]] and the [[Radiometry|radiometric quantities]] used in [[Rendering (computer graphics)|rendering]].

The rendering equation is defined at every point on every surface in the scene being rendered, including points hidden from the camera. The incoming light quantities on the right side of the equation usually come from the left (outgoing) side at other points in the scene ([[ray casting]] can be used to find these other points). The [[Radiosity (computer graphics)|radiosity]] rendering method solves a [[Discretization|discrete approximation]] of this system of equations.<ref name="Immel1986"/> In [[distributed ray tracing]], the integral on the right side of the equation may be evaluated using [[Monte Carlo integration]] by randomly sampling possible incoming light directions. [[Path tracing]] improves and simplifies this method.<ref name="Kajiya1986"/>

The rendering equation can be extended to handle effects such as [[fluorescence]] (in which some absorbed energy is re-emitted at different wavelengths) and can support  transparent and translucent materials by using a [[Bidirectional scattering distribution function|bidirectional scattering distribution function (BSDF)]] in place of a BRDF.<ref name="Glassner95">{{cite book
    | last1 = Glassner
    | first1 = Andrew S.
    | author-link = Andrew Glassner
    | date = 2011
    | orig-date = 1995
    | title = Principles of digital image synthesis
    | publisher = Morgan Kaufmann Publishers, Inc.
    | isbn = 978-1-55860-276-2
    | url = https://www.realtimerendering.com/Principles_of_Digital_Image_Synthesis_v1.0.1.pdf
    | version = 1.0.1
    | access-date = 2025-05-15
    | page=722
    }}</ref> The theory of path tracing sometimes uses a ''path integral'' (integral over possible paths from a light source to a point) instead of the integral over possible incoming directions.<ref name="Veach1997">{{cite thesis
    | last = Veach
    | first = Eric
    | title = Robust Monte Carlo methods for light transport simulation
    | year = 1997
    | degree = PhD
    | publisher = Stanford University
    | url = https://graphics.stanford.edu/papers/veach_thesis/thesis.pdf
    | access-date = 2025-05-15
    | pages=219-220
    }}</ref>