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Interval Analysis For Computer Graphics
 Computer Graphics
, 1992
"... This paper discusses how interval analysis can be used to solve a wide variety of problems in computer graphics. These problems include ray tracing, interference detection, polygonal decomposition of parametric surfaces, and CSG on solids bounded by parametric surfaces. Only two basic algorithms are ..."
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Cited by 154 (2 self)
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This paper discusses how interval analysis can be used to solve a wide variety of problems in computer graphics. These problems include ray tracing, interference detection, polygonal decomposition of parametric surfaces, and CSG on solids bounded by parametric surfaces. Only two basic algorithms are required: SOLVE, which computes solutions to a system of constraints, and MINIMIZE, which computes the global minimum of a function, subject to a system of constraints. We present algorithms for SOLVE and MINIMIZE using interval analysis as the conceptual framework. Crucial to the technique is the creation of "inclusion functions" for each constraint and function to be minimized. Inclusion functions compute a bound on the range of a function, given a similar bound on its domain, allowing a branch and bound approach to constraint solution and constrained minimization. Inclusion functions also allow the MINIMIZE algorithm to compute global rather than local minima, unlike many other numerica...
Simplex and Diamond Hierarchies: Models and Applications
, 2010
"... Hierarchical spatial decompositions are a basic modeling tool in a variety of application domains. Several papers on this subject deal with hierarchical simplicial decompositions generated through simplex bisection. Such decompositions, originally developed for finite elements, are extensively used ..."
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Cited by 11 (4 self)
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Hierarchical spatial decompositions are a basic modeling tool in a variety of application domains. Several papers on this subject deal with hierarchical simplicial decompositions generated through simplex bisection. Such decompositions, originally developed for finite elements, are extensively used as the basis for multiresolution models of scalar fields, such as terrains, and static or timevarying volume data. They have also been used as an alternative to quadtrees and octrees as spatial access structures and in other applications. In this state of the art report, we distinguish between approaches that focus on a specific dimension and those that apply to all dimensions. The primary distinction among all such approaches is whether they treat the simplex or clusters of simplexes, called diamonds, as the modeling primitive. This leads to two classes of data structures and to different query approaches. We present the hierarchical models in a dimension–independent manner, and organize the description of the various applications, primarily interactive terrain rendering and isosurface extraction, according to the dimension of the domain.
SpaceTime Bounds for Collision Detection
, 1993
"... Collision detection and response is an important but costly component of computer animation. We identify three basic reasons why collisiondetection algorithms can be slow. We present a new collisiondetection algorithm that directly addresses two of these problems and that prepares us for future wor ..."
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Cited by 8 (1 self)
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Collision detection and response is an important but costly component of computer animation. We identify three basic reasons why collisiondetection algorithms can be slow. We present a new collisiondetection algorithm that directly addresses two of these problems and that prepares us for future work on the third. Our algorithm is based on a fourdimensional structure that puts a conservative bound on motion through time, even if that motion is specified interactively by a user. The empirical results we present suggest that our algorithm performs significantly better than previous algorithms.
Triangulation and Display of Rational Parametric Surfaces
, 1994
"... We present a comprehensive algorithm to construct a topologically correct triangulation of the real affine part of a rational parametric surface with few restrictions on the defining rational functions. The rational functions are allowed to be undefined on domain curves ( pole curves) and at certain ..."
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Cited by 8 (2 self)
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We present a comprehensive algorithm to construct a topologically correct triangulation of the real affine part of a rational parametric surface with few restrictions on the defining rational functions. The rational functions are allowed to be undefined on domain curves ( pole curves) and at certain special points (base points), and the surface is allowed to have nodal or cuspidal selfintersections. We also recognize that for a complete display some real points on the parametric surface may be generated only by complex parameter values, and that some finite points on the surface may be generated only by infinite parameter values; we show how to compensate for these conditions. Our techniques for handling these problems have applications in scientific visualization, rendering nonstandard NURBS, and in finiteelement mesh generation.
Surface Modeling Using Quadtrees
, 1996
"... Two quadtree variants effective in modeling 2 1 2 D surfaces are presented. The restricted quadtree can handle regularly sampled data. For irregular data, embedding a TIN inside a PMR quadtree is suggested. Together, these schemes facilitate the handling of most types of input within a single fra ..."
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Cited by 6 (0 self)
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Two quadtree variants effective in modeling 2 1 2 D surfaces are presented. The restricted quadtree can handle regularly sampled data. For irregular data, embedding a TIN inside a PMR quadtree is suggested. Together, these schemes facilitate the handling of most types of input within a single framework. Algorithms for the construction of both data structures from their respective data formats are described and analyzed. The possible application of each of the models to the problem of visibility determination is considered and its performance is theoretically evaluated. The support of the Department of Energy under Grant DEFG0295ER25237 and the National Science Foundation under Grant IRI9216970 is gratefully acknowledged, as is the help of Sandy German in preparing this paper. Preface This dissertation deals with modeling of surfaces, i.e., data structures that can facilitate the storage and manipulation of objects resembling topographical surfaces using computers. Such surfac...
unknown title
"... This paper discusses how interval analysis can be used to solve a wide variety of problems in computer graphics. These problems include ray tracing, interference detection, polygonal decomposition of parametric surfaces, and CSG on solids bounded by parametric surfaces. Only two basic algorithms are ..."
Abstract
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This paper discusses how interval analysis can be used to solve a wide variety of problems in computer graphics. These problems include ray tracing, interference detection, polygonal decomposition of parametric surfaces, and CSG on solids bounded by parametric surfaces. Only two basic algorithms are required: SOLVE, which computes solutions to a system of constraints, and MINIMIZE, which computes the global minimum of a function, subject to a system of constraints. We present algorithms for SOLVE and MINIMIZE using interval analysis as the conceptual framework. Crucial to the technique is the creation of “inclusion functions ” for each constraint and function to be minimized. Inclusion functions compute a bound on the range of a function, given a similar bound on its domain, allowing a branch and bound approach to constraint solution and constrained minimization. Inclusion functions also allow the MINIMIZE algorithm to compute global rather than local minima, unlike many other numerical algorithms. Some very recent theoretical results are presented regarding existence and uniqueness of roots of nonlinear equations, and global parameterizability of implicitly described manifolds. To illustrate the power of the approach, the basic algorithms are further developed into a new algorithm for the approximation of implicit curves.