Annex: Mirror BRDF - Patrick Bell

This annex to Derivation of Physically Based Rendering demonstrates energy conservation for a mirror BRDF.

Define

where $k\in[0,1]$ . Then we need

The delta function “transforms” the integral to a point where $\theta_i = \theta_r$ and $\phi_i = \phi_r - \pi$ . So

because $\phi_i \in [0,\pi/2]$ . The mirror BRDF is energy conserving.