Feature Aligned Volume Manipulation for Illustration and Visualization
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 <u><i>l</i></u> = <u><i>at</i></u> + <u><i>b</i></u>, <i>t</i> &gt; 0 with a parametric surface <u><i>x</i></u> = <i><u>f</u>(u, v)</i>, and with implicit surfaces <i>f(x,y,z)</i> = 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 <i>u, v</i>, and <i>t</i>. 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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