A unique graphics pipeline and method for utilizing the same to render trimmed non-uniform rational b-spline (NURBS) surfaces is described. To render a graphics primitive such as a trimmed NURBS, the primitive first is processed through a compilation step, followed by a two phase traversal step and a display step. Through this process, the NURBS surface is reduced at creation time into a form amenable for fast processing subsequently during traversal. At compilation the trimmed NURBS is broken down into monotone v-regions which remain valid for subsequent views and renderings. The form produced by compilation is independent of the tessellation step size; therefore the form remains valid under heavy changes to modeling and/or viewing transformations. Furthermore, a substantial portion of the processing complexity for rendering the primitive is absorbed at compilation, thus minimizing the complexity of tessellation at traversal. Once the graphic primitive is processed through the compilation step, the primitive is processed through the two phase traversal step. The two phases may be processed in parallel.
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