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Realtime GPU rendering of piecewise algebraic surfaces
 In SIGGRAPH ’06: ACM SIGGRAPH 2006 Papers
"... Figure 1: Several states of an animated fourth order algebraic surface rendered in realtime using our technique. We consider the problem of realtime GPU rendering of algebraic surfaces defined by Bézier tetrahedra. These surfaces are rendered directly in terms of their polynomial representations, ..."
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Cited by 51 (0 self)
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Figure 1: Several states of an animated fourth order algebraic surface rendered in realtime using our technique. We consider the problem of realtime GPU rendering of algebraic surfaces defined by Bézier 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.
C¹ Modeling with APatches from Rational Trivariate Functions
, 2001
"... We approximate a manifold triangulation in R³ using smooth implicit algebraic surface patches, which we call Apatches. Here each Apatch is a real isocontour of a trivariate rational function defined within a tetrahedron. The rational trivariate function provides increased degrees of freedom so t ..."
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Cited by 9 (4 self)
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We approximate a manifold triangulation in R³ using smooth implicit algebraic surface patches, which we call Apatches. Here each Apatch is a real isocontour of a trivariate rational function defined within a tetrahedron. The rational trivariate function provides increased degrees of freedom so that the number of surface patches needed for freeform shape modeling is significantly reduced compared to earlier similar approaches. Furthermore, the surface patches have quadratic precision, that is they exactly recover quadratic surfaces. We give conditions under which a C¹ smooth and single sheeted surface patch is isolated from the multiple sheets.
ASplines: Local Interpolation and Approximation using G^kContinuous Piecewise Real Algebraic Curves
"... We provide sufficient conditions for the BernsteinBezier (BB) form of an implicitly defined bivariate polynomial over a triangle, such that the zero contour of the polynomial defines a smooth and single sheeted real algebraic curve segment. We call a piecewise G^kcontinuous chain of such real alge ..."
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Cited by 5 (1 self)
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We provide sufficient conditions for the BernsteinBezier (BB) form of an implicitly defined bivariate polynomial over a triangle, such that the zero contour of the polynomial defines a smooth and single sheeted real algebraic curve segment. We call a piecewise G^kcontinuous chain of such real algebraic curve segments in BBform as an Aspline (short for algebraic spline). We prove that the degree n Asplines can achieve in general G 2n,3 continuity by local fitting and still have degrees of freedom to achieve local data approximation. As examples, we show how to construct locally convex cubic Asplines to interpolate and/or approximate the vertices of an arbitrary planar polygon with up to G 4 continuity, to fit discrete points and derivatives data, and approximate high degree parametric and implicitly de ned curves. Additionally, we provide computable error bounds.
Abstract RealTime GPU Rendering of Piecewise Algebraic Surfaces
"... Figure 1: Several states of an animated fourth order algebraic surface rendered in realtime using our technique. We consider the problem of realtime GPU rendering of algebraic surfaces defined by Bézier tetrahedra. These surfaces are rendered directly in terms of their polynomial representations, ..."
Abstract
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Figure 1: Several states of an animated fourth order algebraic surface rendered in realtime using our technique. We consider the problem of realtime GPU rendering of algebraic surfaces defined by Bézier 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.
Trivariate Functions Institute of Computational Mathematics, Chinese Academy of Sciences Abstract
"... We approximate a manifold triangulation in IR 3 using smooth implicit algebraic surface patches, which we call Apatches. Here each Apatch is a real isocontour of a trivariate rational function de ned within a tetrahedron. The rational trivariate function provides increased degrees of freedom so t ..."
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We approximate a manifold triangulation in IR 3 using smooth implicit algebraic surface patches, which we call Apatches. Here each Apatch is a real isocontour of a trivariate rational function de ned within a tetrahedron. The rational trivariate function provides increased degrees of freedom so that the number of surface patches needed for freeform shape modeling is signi cantly reduced compared to earlier similar approaches. Furthermore, the surface patches have quadratic precision, that is they exactly recover quadratic surfaces. We give conditions under which aC 1 smooth and single sheeted surface patch is isolated from the multiple sheets. Key words: Algebraic surface; rational Apatch; surface t; triangulation 1
The Design and Implementation of a Programming Infrastructure for the Integration and Application of Implicit Surface Research
"... We describe a C++ class hierarchy to support the efficient implementation of implicit surface functions and algorithms. The structure of the hierarchy allows many methods to be defined automatically over a wide range of specific implicit surface types. This structure also provides the infrasture for ..."
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We describe a C++ class hierarchy to support the efficient implementation of implicit surface functions and algorithms. The structure of the hierarchy allows many methods to be defined automatically over a wide range of specific implicit surface types. This structure also provides the infrasture for integrating many various methods and models found in modern implicit surface research. This paper also contains helpful derivations for automatically maintaining derivates, interval extensions and Lipschitz bounds of functions as they are composed from simpler elements. An object specifically designed to support blending simplifies the implementation of new and existing blending techniques. Sample implementations of some members of the library are provided as examples to better understand the details of the implementation. We expect this work to result in a tool the implicit surface community finds valuable, and provide an effective way for this community to share the efficient implementation of its results. 1
Modeling with Hybrid Multiplesided Apatches
"... We propose a new scheme for modeling a smooth interpolatory surface, from a surface discretization consisting of triangles, quadrilaterals and pentagons, by algebraic surface patches which are subsets of real zero contours of trivariate rational functions defined on a collection of tetrahedra and ..."
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We propose a new scheme for modeling a smooth interpolatory surface, from a surface discretization consisting of triangles, quadrilaterals and pentagons, by algebraic surface patches which are subsets of real zero contours of trivariate rational functions defined on a collection of tetrahedra and pyramids. The rational form of the modeling function provides enough degrees of freedom so that the number of the surface patches is significantly reduced, and the surface has quadratic recover property.