Past exam of the computer science course of the University of Oxford / 2015 / Computer graphics / 3 / a / Solution

At a surface point $p$, let $\hat{N}$ be the outward unit normal vector, $\hat{V}$ the unit vector from $p$ toward the viewer, and $\hat{L}$ the unit vector from $p$ toward the light. The incoming propagation direction is therefore $-\hat{L}$. The specular reflection direction $\hat{R}$ points away from the surface along the ideal reflected ray.

Resolve $\hat{L}$ into its normal and tangential components:Reflection of the incoming direction preserves its tangential component and reverses its normal component. Equivalently, the outgoing direction has the same normal component as $\hat{L}$ and the opposite tangential component, givingThe dot product identity shows $\Vert \hat{R}\Vert =1$ when $\Vert \hat{L}\Vert =\Vert \hat{N}\Vert =1$.

In the Phong reflection model, ambient lighting approximates unmodelled indirect illumination and is independent of these directions. The diffuse term models Lambertian reflectance and varies as $\max (0,\hat{N}\cdot \hat{L})$. The specular term models a highlight around $\hat{R}$ and varies as $\max (0,\hat{R}\cdot \hat{V}{)}^n$, where $n>0$ controls its sharpness. For a visible front surface, a convenient per-colour-channel equation iswhere $I_a$ is ambient illumination, $I_L$ is the source strength at the surface including any chosen attenuation, $k_a,k_d,k_s\ge 0$ are material weights, and $\chi$ is one when $\hat{N}\cdot \hat{L}>0$ and zero otherwise. The weights can be colour-dependent. An idealized point source's incident strength decreases as inverse distance squared; the empirical equation can also use a prescribed attenuation law.

In degenerate geometric cases the effective weights switch off. A light behind or tangent to the surface contributes neither direct diffuse nor specular illumination. If $\hat{R}\cdot \hat{V}\le 0$, the specular term is zero. A viewpoint on the back side is rejected by back-face culling for a one-sided surface. A material with no highlight has $k_s=0$; with no diffuse response, $k_d=0$; without ambient illumination the ambient term vanishes. These zeros do not require renormalizing the other weights, and the empirical Phong reflection model does not impose a universal condition $k_a+k_d+k_s=1$. At coincident light/surface or eye/surface positions a direction cannot be normalized; such singular configurations require a separate convention, rather than substituting an undefined unit vector into the formula.