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143
Robust Epsilon Visibility
 SIGGRAPH
, 2002
"... Analytic visibility algorithms, for example methods which compute a subdivided mesh to represent shadows, are notoriously unrobust and hard to use in practice. We present a new method based on a generalized definition of extremal stabbing lines, which are the extremities of shadow boundaries. We tre ..."
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Cited by 40 (1 self)
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Analytic visibility algorithms, for example methods which compute a subdivided mesh to represent shadows, are notoriously unrobust and hard to use in practice. We present a new method based on a generalized definition of extremal stabbing lines, which are the extremities of shadow boundaries. We treat scenes containing multiple edges or vertices in degenerate configurations, (e.g., collinear or coplanar). We introduce a robust ɛ method to determine whether each generalized extremal stabbing line is blocked, or is touched by these scene elements, and thus added to the line's generators. We develop robust blocker predicates for polygons which are smaller than ɛ. For larger ɛ values, small shadow features merge and eventually disappear. We can thus robustly connect generalized extremal stabbing lines in degenerate scenes to form shadow boundaries. We show that our approach is consistent, and that shadow boundary connectivity is preserved when features merge. We have implemented our algorithm, and show that we can robustly compute analytic shadow boundaries to the precision of our chosen ɛ threshold for nontrivial models, containing numerous degeneracies.
A Fortran 90 Environment for Research and Prototyping of Enclosure Algorithms for Nonlinear Equations and Global Optimization
"... An environment for general research into and prototyping of algorithms for reliable constrained and unconstrained global nonlinear optimization and reliable enclosure of all roots of nonlinear systems of equations, with or without inequality constraints, is being developed. This environment should b ..."
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Cited by 40 (19 self)
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An environment for general research into and prototyping of algorithms for reliable constrained and unconstrained global nonlinear optimization and reliable enclosure of all roots of nonlinear systems of equations, with or without inequality constraints, is being developed. This environment should be portable, easy to learn, use, and maintain, and sufficiently fast for some production work. The motivation, design principles, uses, and capabilities for this environment are outlined. The environment includes an interval data type, a symbolic form of automatic differentiation to obtain an internal representation for functions, a special technique to allow conditional branches with operator overloading and interval computations, and generic routines to give interval and noninterval function and derivative information. Some of these generic routines use a special version of the backward mode of automatic differentiation. The package also includes dynamic data structures for exhaustive sear...
Adaptive implicit surface polygonization using marching triangles
 COMPUTER GRAPHICS FORUM
, 2001
"... This paper presents several improvements to the marching triangles algorithm for general implicit surfaces. The original method generates equilateral triangles of constant size almost everywhere on the surface. We present several modifications to adapt the size of the triangles to the curvature of t ..."
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Cited by 39 (6 self)
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This paper presents several improvements to the marching triangles algorithm for general implicit surfaces. The original method generates equilateral triangles of constant size almost everywhere on the surface. We present several modifications to adapt the size of the triangles to the curvature of the surface. As cracks may arise in the resulting polygonization, we propose a specific crackclosing method invoked at the end of the mesh growing step. Eventually, we show that the marching triangles can be used as an incremental meshing technique in an interactive modeling environment. In contrast to existing incremental techniques based on spatial sudvision, no extra datastructure is needed to incrementally edit skeletal implicit surfaces, which saves both memory and computation time.
SelfValidated Numerical Methods and Applications
, 1997
"... erical methods. We apologize to the reader for the length and verbosity of these notes but, like Pascal, 1 we didn't have the time to make them shorter. 1 "Je n'ai fait celleci plus longue que parce que je n'ai pas eu le loisir de la faire plus courte." Blaise Pascal, ..."
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Cited by 33 (0 self)
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erical methods. We apologize to the reader for the length and verbosity of these notes but, like Pascal, 1 we didn't have the time to make them shorter. 1 "Je n'ai fait celleci plus longue que parce que je n'ai pas eu le loisir de la faire plus courte." Blaise Pascal, Lettres Provinciales, XVI (1657). i ii Acknowledgements We thank the Organizing Committee of the 21 st Brazilian Mathematics Colloquium for the opportunity to present this course. We wish to thank Jo~ao Comba, who helped implement a prototype affine arithmetic package in Modula3, and Marcus Vinicius Andrade, who helped debug the C version and wrote an implicit surface raytracer based on it. Ronald van Iwaarden contributed an independent implementation of AA, and investigated its performance on branchandbound global optimization algorithms. Douglas Priest and Helmut Jarausch provided code and advice for rounding mode control. W
Sampling of Procedural Shaders Using Affine Arithmetic
, 1996
"... Procedural shaders have become popular tools for describing surface reflectance functions and other material properties. In comparison to fixed resolution textures they have the advantage of being resolution independent and storage e#cient. While procedural shaders provide an interface for evaluati ..."
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Cited by 33 (5 self)
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Procedural shaders have become popular tools for describing surface reflectance functions and other material properties. In comparison to fixed resolution textures they have the advantage of being resolution independent and storage e#cient. While procedural shaders provide an interface for evaluating the shader at a single point in parameter space, it is not easily possible to obtain an average value of the shader together with accurate error bounds over a finite area. Yet the ability to compute such error bounds is crucial for several interesting applications, most notably hierarchical area sampling for global illumination computations using the finite element approach and for the generation of textures used in interactive computer graphics. Using a#ne arithmetic for evaluating the shader over a finite area yields a tight, conservative error interval for the shader function. Compilers can automatically generate code for utilizing a#ne arithmetic from within shaders implemented in a ...
Adaptive Enumeration of Implicit Surfaces with Affine Arithmetic
 Computer Graphics Forum
, 1996
"... . We discuss adaptive enumeration and rendering methods for implicit surfaces, using octrees computed with affine arithmetic, a new tool for range analysis. Affine arithmetic is similar to standard interval arithmetic, but takes into account correlations between operands and subformulas, generally ..."
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Cited by 32 (15 self)
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. We discuss adaptive enumeration and rendering methods for implicit surfaces, using octrees computed with affine arithmetic, a new tool for range analysis. Affine arithmetic is similar to standard interval arithmetic, but takes into account correlations between operands and subformulas, generally providing much tighter bounds for the computed quantities. The resulting octrees are accordingly much smaller, and the rendering faster. We also describe applications of affine arithmetic to intersection and ray tracing of implicit surfaces. keywords: cellular models, interval analysis, rendering, implicit surfaces. 1 Introduction Implicit surfaces have recently become popular in computer graphics and solid modeling. In order to exploit existing hardware and algorithms, it is often necessary to approximate such surfaces by models with simpler geometry, such as polygonal meshes or voxel arrays. Let S be a surface defined implicitly by the equation h(x; y; z) = 0. A simple and general techn...
Reduction Of Constraint Systems
, 1993
"... Geometric modeling by constraints leads to large systems of algebraic equations. This paper studies bipartite graphs underlaid by systems of equations. It shows how these graphs make possible to polynomially decompose these systems into well constrained, over, and underconstrained subsystems. This ..."
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Cited by 30 (2 self)
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Geometric modeling by constraints leads to large systems of algebraic equations. This paper studies bipartite graphs underlaid by systems of equations. It shows how these graphs make possible to polynomially decompose these systems into well constrained, over, and underconstrained subsystems. This paper also gives an efficient method to decompose well constrained systems into irreducible ones. These decompositions greatly speed up the resolution in case of reducible systems. They also allow debugging systems of constraints. Key Words: geometric modeling, constraints, bipartite graphs, matching, maximum matching, perfect matching. 1. INTRODUCTION Geometric modeling by constraints is an interesting approach in CAD. Typically, in 2D, geometric modeling by constraints specifies geometrical objects such as points, lines, circles, conics by a set of constraints : distances between points, points and lines, parallel lines, angles between lines, incidence relations between points and lines,...
Generative Modeling: A Symbolic System for Geometric Modeling
 Computer Graphics
, 1992
"... This paper discusses a new, symbolic approach to geometric modeling called generative modeling. The approach allows specification, rendering, and analysis of a wide variety of shapes including 3D curves, surfaces, and solids, as well as higherdimensional shapes such as surfaces deforming in time, a ..."
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Cited by 30 (1 self)
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This paper discusses a new, symbolic approach to geometric modeling called generative modeling. The approach allows specification, rendering, and analysis of a wide variety of shapes including 3D curves, surfaces, and solids, as well as higherdimensional shapes such as surfaces deforming in time, and volumes with a spatially varying mass density. The system also supports powerful operations on shapes such as "reparameterize this curve by arclength", "compute the volume, center of mass, and moments of inertia of the solid bounded by these surfaces", or "solve this constraint or ODE system". The system has been used for a wide variety of applications, including creating surfaces for computer graphics animations, modeling the fur and body shape of a teddy bear, constructing 3D solid models of elastic bodies, and extracting surfaces from magnetic resonance (MR) data. Shapes in the system are specified using a language which builds multidimensional parametric functions. The language is bas...
Raytracing Procedural Displacement Shaders
 In Graphics Interface
, 1998
"... Displacement maps and procedural displacement shaders are a widely used approach of specifying geometric detail and increasing the visual complexity of a scene. While it is relatively straightforward to handle displacement shaders in pipeline based rendering systems such as the Reyesarchitecture, i ..."
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Cited by 29 (1 self)
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Displacement maps and procedural displacement shaders are a widely used approach of specifying geometric detail and increasing the visual complexity of a scene. While it is relatively straightforward to handle displacement shaders in pipeline based rendering systems such as the Reyesarchitecture, it is much harder to efficiently integrate displacementmapped surfaces in raytracers. Many commercial raytracers tessellate the surface into a multitude of small triangles. This introduces a series of problems such as excessive memory consumption and possibly undetected surface detail. In this paper we describe a novel way of raytracing procedural displacement shaders directly, that is, without introducing intermediate geometry. Affine arithmetic is used to compute bounding boxes for the shader over any range in the parameter domain. The method is comparable to the direct raytracing of B'ezier surfaces and implicit surfaces using B'ezier clipping and interval methods, respectively. Keyw...
Functional Implementations of Continuous Modeled Animation (Expanded Version)
, 1998
"... Animation is a temporally continuous phenomenon, but is typically programmed in terms of a discrete sequence of changes. The use of discreteness serves to accommodate the machine that is presenting an animation, rather than the person modeling an animation with the help of a computer. Using a co ..."
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Cited by 27 (6 self)
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Animation is a temporally continuous phenomenon, but is typically programmed in terms of a discrete sequence of changes. The use of discreteness serves to accommodate the machine that is presenting an animation, rather than the person modeling an animation with the help of a computer. Using a continuous model of time for animation allows for natural specification, avoiding some artificial details, but is difficult to implement with generality, robustness and efficiency. This paper presents and motivates continuous modeled animation, and sketches out a naive functional implementation for it. An examination of some of the practical problems with this implementation leads to several alternate representations, all of which have difficulties in themselves, some quite subtle. We hope that the insights and techniques discussed in this paper lead to still better representations, so that animation may be specified in natural terms without significant loss of performance.