Fast Phong shading

Fast Phong shading is an approximation of Phong shading that avoids the expensive per-pixel normalisation and power calculation.

In normal Phong shading, the interpolated normal N must be normalised at every pixel (a square root and a division), and (N·H)^n must be computed. Fast Phong shading reduces this work.

Idea. Over a triangle, the interpolated (unnormalised) normal is a linear function of the screen position:

N = Ax + By + C

where the vectors A, B, C are found from the three vertex normals. For a distant light (L constant), the diffuse term becomes

I_diff = (N·L)/(|L||N|) = (ax + by + c) / √(dx² + exy + fy² + gx + hy + i)

where the constants a, b, …, i stand for the various dot products of A, B, C and L. The method then uses:

1. Taylor series expansion of this expression, keeping terms up to the second degree in x and y. This removes the per-pixel square root and division:

   I_diff ≈ T5·x² + T4·xy + T3·y² + T2·x + T1·y + T0

   The specular term is approximated in the same way.

2. Incremental (forward-difference) evaluation: along a scan line, I(x + 1) = I(x) + ΔI and ΔI(x + 1) = ΔI(x) + Δ²I, so each pixel costs only two additions.

Advantages over Phong shading

• Much faster, with no per-pixel square root, division or exponentiation. Fast Phong takes about twice as long as Gouraud shading, while normal Phong takes about six to seven times as long.
• Nearly the same image quality (good highlights).
• Less computation, so it is suitable for hardware.

Limitation: approximation errors can appear if the polygon is large or the normals change quickly across it (especially for large n).