Real-time GPU rendering of piecewise algebraic surfaces

Charles Loop, Jim Blinn
Microsoft Research
In SIGGRAPH ’06: ACM SIGGRAPH 2006 Papers (2006), pp. 664-670


   title={Real-time GPU rendering of piecewise algebraic surfaces},

   author={Loop, C.T. and Blinn, J.F.},



   publisher={Google Patents},

   note={US Patent App. 11/495,246}


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We consider the problem of real-time GPU rendering of algebraic surfaces defined by Bezier tetrahedra. These surfaces are rendered directly in terms of their polynomial representations, as opposed to a collection of approximating triangles, thereby eliminating tessellation artifacts and reducing memory usage. A key step in such algorithms is the computation of univariate polynomial coefficients at each pixel; real roots of this polynomial correspond to possibly visible points on the surface. Our approach leverages the strengths of GPU computation and is highly efficient. Furthermore, we compute these coefficients in Bernstein form to maximize the stability of root finding, and to provide shader instances with an early exit test based on the sign of these coefficients. Solving for roots is done using analytic techniques that map well to a SIMD architecture, but limits us to fourth order algebraic surfaces. The general framework could be extended to higher order with numerical root finding.
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