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====== Bidirectional Reflectance Distribution Function ======
The **bidirectional reflectance distribution function (BRDF)** is a function that defines how <color #fff200>light</color> from a source is reflected off an opaque surface.

==== Notation ====
Let <color #ff7f27>$\displaystyle f_r(\omega_i,\omega_r)$</color> be the BRDF where:

<color #00a2e8>$\omega$</color> is direction <color #00a2e8>$(\theta,\phi)$</color>

<color #00a2e8>$i$</color> is irradiance

<color #00a2e8>$r$</color> is radiance

<color #ff7f27>$\displaystyle f_r(\omega_i,\omega_r) = \frac{\mathrm{d}L(\omega_r)}{\mathrm{d}E(\omega_i)}$</color>

==== Properties of physically based BRDFs ====

__Positivity__

<color #ffc90e>$f_r(\omega_i,\omega_r) \geq 0$</color>

__Helmholtz reciprocity__

<color #ffc90e>$f_r(\omega_i,\omega_r) = f_r(\omega_r,\omega_i)$</color>

__Energy conservation__

<color #ffc90e>$\displaystyle \forall \omega_i, \int_{\Omega} f_r(\omega_i,\omega_r) \cos \theta_r \mathrm{d} \omega_r \leq 1$</color>

==== See also ====
  * [[article:computer_graphics|Computer graphics]]
  * [[article:pbr|Physically based rendering]]