Ray tracing deformed surfaces

A collection of new methods for ray tracing differentiable surfaces is developed. The methods are general, and extend the set of "ray-traceable" surfaces suitable for use in geometric modeling. We intersect a ray l = at + b , t > 0 with a parametric surface x = f (u, v) , and with implicit surfaces f(x,y,z) = 0. A smooth surface is treated as a deformation of a flat sheet; the intersection problem is converted to a new coordinate system in which the surfaces are flat, and the rays are bent. We develop methods for providing good initial estimates of the parametric intersection values, and a "closeness criterion," to reduce computation. These same criteria help us substitute a set of simpler surfaces for the more complex surface. The parametric method produces the intersection values of u, v , and t . These are suitable for shading calculations and for mapping textures onto the surface; they can also produce the local coordinate frame values, suitable for anisotropic lighting models.