Ray Tracing of Quadratic Parametric Surface

Abstract

Over the past decades, vast research has been done on the ray-triangle intersect test but not much attention has been paid to the ray-quadratic parametric surface intersection test. In this chapter we present two direct ray tracing methods for quadratic parametric surfaces and introduce a simple optimization technique for them.

Appendix

The geometric normal ${N_G}$ of can be derived as

$$\begin{aligned} {N_G} = \left( \frac{\partial Q(u, v)}{\partial u} \times \frac{\partial Q(u, v)}{\partial v} \right) \bigg / \left| \frac{\partial Q(u, v)}{\partial u} \times \frac{\partial Q(u, v)}{\partial v} \right| . \end{aligned}$$

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The partial derivatives $\frac{\partial Q(u, v)}{\partial u}$ and $\frac{\partial Q(u, v)}{\partial v}$ are obtained as

$$\begin{aligned} \frac{\partial Q(u, v)}{\partial u}&= (2{A} - {E})u + ({D}-{F})v + ({E}-2{C})w\end{aligned}$$

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$$\begin{aligned} \frac{\partial Q(u, v)}{\partial v}&= (2{B} - {F})v + ({D}-{E})u + ({F}-2{C})w. \end{aligned}$$

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For smooth rendering, we use the Phong-interpolated normal ${N_p}$ as in [3] instead of ${N_G}$. The geometric normal ${N_G}$ should be used, for example, when the dot product of ${N_G}$ and a reflected vector computed with ${N_p}$ is negative.