“Ray tracing deformed surfaces” by Barr

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.

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