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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 132 (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...
Guaranteeing the Topology of an Implicit Surface Polygonization for Interactive Modeling
, 1997
"... Morse theory shows how the topology of an implicit surface is affected by its function's critical points, whereas catastrophe theory shows how these critical points behave as the function's parameters change. Interval analysis finds the critical points, and they can also be tracked efficiently durin ..."
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Cited by 99 (9 self)
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Morse theory shows how the topology of an implicit surface is affected by its function's critical points, whereas catastrophe theory shows how these critical points behave as the function's parameters change. Interval analysis finds the critical points, and they can also be tracked efficiently during parameter changes. Changes in the function value at these critical points cause changes in the topology. Techniques for modifying the polygonization to accommodate such changes in topology are given. These techniques are robust enough to guarantee the topology of an implicit surface polygonization, and are efficient enough to maintain this guarantee during interactive modeling. The impact of this work is a topologicallyguaranteed polygonization technique, and the ability to directly and accurately manipulate polygonized implicit surfaces in real time.
Interactive ray tracing of arbitrary implicits with simd interval arithmetic
 In Proceedings of the 2nd IEEE/EG Symposium on Interactive Ray Tracing
, 2007
"... We present a practical and efficient algorithm for interactively ray tracing arbitrary implicit surfaces. We use interval arithmetic (IA) both for robust root computation and guaranteed detection of topological features. In conjunction with ray tracing, this allows for rendering literally any progra ..."
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Cited by 21 (7 self)
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We present a practical and efficient algorithm for interactively ray tracing arbitrary implicit surfaces. We use interval arithmetic (IA) both for robust root computation and guaranteed detection of topological features. In conjunction with ray tracing, this allows for rendering literally any programmable implicit function simply from its definition. Our method requires neither special hardware, nor preprocessing or storage of any data structure. Efficiency is achieved through SIMD optimization of both the interval arithmetic computation and coherent ray traversal algorithm, delivering interactive results even for complex implicit functions.
Polygonizing Implicit Surfaces With Guaranteed Topology
, 1997
"... by Barton Talbot Stander, Ph.D. Washington State University May 1997 Chair: John C. Hart An interactive modeling system for implicit surfaces is presented. The display consists of a polygonal approximation which is guaranteed to have the same topology as the implicit surface. The current work focuse ..."
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Cited by 7 (1 self)
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by Barton Talbot Stander, Ph.D. Washington State University May 1997 Chair: John C. Hart An interactive modeling system for implicit surfaces is presented. The display consists of a polygonal approximation which is guaranteed to have the same topology as the implicit surface. The current work focuses on blended ellipsoids, but could be extended to include any smooth, bounded implicit surface. A polygonization algorithm and an incremental repolygonization algorithm are provided. Treating an implicit surface as a gradient system allows theorems from Morse theory to describe implicit surface topology. An implicit surface changes topology only when a critical value of its defining function changes sign. These critical points may be found using interval analysis. Techniques for modifying the polygonization to accommodate such changes in topology are given. iv Contents 1 Introduction 1 1.1 Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Implicit Surfaces...
Robust Adaptive Approximation of Implicit Curves
, 2001
"... . We present an algorithm for computing a robust adaptive polygonal approximation of an implicit curve in the plane. The approximation is adapted to the geometry of the curve because the length of the edges varies with the curvature of the curve. Robustness is achieved by combining interval arithm ..."
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Cited by 3 (1 self)
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. We present an algorithm for computing a robust adaptive polygonal approximation of an implicit curve in the plane. The approximation is adapted to the geometry of the curve because the length of the edges varies with the curvature of the curve. Robustness is achieved by combining interval arithmetic and automatic differentiation. Keywords: piecewise linear approximation; interval arithmetic; automatic differentiation; geometric modeling. 1
Robust Adaptive Polygonal Approximation of Implicit Curves
"... We present an algorithm for computing a robust adaptive polygonal approximation of an implicit curve in the plane. The approximation is adapted to the geometry of the curve because the length of the edges varies with the curvature of the curve. Robustness is achieved by combining interval arithmet ..."
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Cited by 1 (0 self)
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We present an algorithm for computing a robust adaptive polygonal approximation of an implicit curve in the plane. The approximation is adapted to the geometry of the curve because the length of the edges varies with the curvature of the curve. Robustness is achieved by combining interval arithmetic and automatic differentiation.
ContourBased Polygonization of Regular Grid Terrain Data
"... Terrain visualization has been an interesting issue for graphics people for long, and there have been a lot of attempts in various ways for more e#cient visualization. In most of the works the terrain data was considered as a triangular mesh as it is, or as an implicit surface in which the original ..."
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Terrain visualization has been an interesting issue for graphics people for long, and there have been a lot of attempts in various ways for more e#cient visualization. In most of the works the terrain data was considered as a triangular mesh as it is, or as an implicit surface in which the original shape#topology# of the terrain can be slightly changed.