“Shading bicubic patches” by Shantz and Lien

We present several techniques for implementing Phong shading in hardware for bicubic patches. Patches are shaded, not by subdividing into polygons, but by drawing many curves close together leaving no pixel gaps. Each curve is drawn using an adaptive forward difference algorithm which generates the coordinates as well as the shading parameters as cubic functions incrementally evaluated along the curve. The forward difference step size is adaptively adjusted so that it generates approximately one pixel along the curve per forward difference step. The hardware implements Phong shading directly with a surprisingly simple configuration built from general purpose compute units and look-up tables. Two new methods are presented for deriving bicubic approximations to the shading parameters over a bicubic patch. One method uses two Coons patches to approximate the unnormalized N·L, and N·H, and a third Coons patch for N·N, where N is the surface normal, L is the light direction, and H is the direction of maximum highlight. In this case the hardware performs the normalization per pixel. The second method uses two Coons patches to approximate the normalized dot products N·L, and N·H. The method is suitable for both hardware and software implementations.

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