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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 efficie ..."
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Cited by 112 (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.
Interval Computations: Introduction, Uses, and Resources
 Euromath Bulletin
, 1996
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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...
Enhanced Interval Analysis for Phase Stability: Cubic Equation of State Models
 IND. ENG. CHEM. RES
, 1998
"... The reliable prediction of phase stability is a challenging computational problem in chemical process simulation, optimization and design. The phase stability problem can be formulated either as a minimization problem or as an equivalent nonlinear equation solving problem. Conventional solution meth ..."
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Cited by 32 (21 self)
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The reliable prediction of phase stability is a challenging computational problem in chemical process simulation, optimization and design. The phase stability problem can be formulated either as a minimization problem or as an equivalent nonlinear equation solving problem. Conventional solution methods are initialization dependent, and may fail by converging to trivial or nonphysical solutions or to a point that is a local but not global minimum. Thus there has been considerable recent interest in developing more reliable techniques for stability analysis. Recently we have demonstrated, using cubic equation of state models, a technique that can solve the phase stability problem with complete reliability. The technique, which is based on interval analysis, is initialization independent, and if properly implemented provides a mathematical guarantee that the correct solution to the phase stability problem has been found. However, there is much room for improvement in the computational efficiency of the technique. In this paper we consider two means of enhancing the efficiency of the method, both based on sharpening the range of interval function evaluations. Results indicate that by using the enhanced method, computation times can be reduced by nearly an order of magnitude in some cases.
Comparison of Interval Methods for Plotting Algebraic Curves
 Comput. Aided Geom. Des
, 2002
"... This paper compares the performance and e#ciency of di#erent function range interval methods for plotting f(x, y) = 0 on a rectangular region based on a subdivision scheme, where f(x, y) is a polynomial. The solution of this problem has many applications in CAGD. ..."
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Cited by 26 (2 self)
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This paper compares the performance and e#ciency of di#erent function range interval methods for plotting f(x, y) = 0 on a rectangular region based on a subdivision scheme, where f(x, y) is a polynomial. The solution of this problem has many applications in CAGD.
Efficient and safe global constraints for handling numerical constraint systems
 SIAM J. NUMER. ANAL
, 2005
"... Numerical constraint systems are often handled by branch and prune algorithms that combine splitting techniques, local consistencies, and interval methods. This paper first recalls the principles of Quad, a global constraint that works on a tight and safe linear relaxation of quadratic subsystems ..."
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Cited by 26 (9 self)
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Numerical constraint systems are often handled by branch and prune algorithms that combine splitting techniques, local consistencies, and interval methods. This paper first recalls the principles of Quad, a global constraint that works on a tight and safe linear relaxation of quadratic subsystems of constraints. Then, it introduces a generalization of Quad to polynomial constraint systems. It also introduces a method to get safe linear relaxations and shows how to compute safe bounds of the variables of the linear constraint system. Different linearization techniques are investigated to limit the number of generated constraints. QuadSolver, a new branch and prune algorithm that combines Quad, local consistencies, and interval methods, is introduced. QuadSolver has been evaluated on a variety of benchmarks from kinematics, mechanics, and robotics. On these benchmarks, it outperforms classical interval methods as well as constraint satisfaction problem solvers and it compares well with stateoftheart optimization solvers.
The Cluster Problem In Multivariate Global Optimization
 Journal of Global Optimization
, 1994
"... . We consider branch and bound methods for enclosing all unconstrained global minimizers of a nonconvex nonlinear twicecontinuously differentiable objective function. In particular, we consider bounds obtained with interval arithmetic, with the "midpoint test," but no acceleration procedu ..."
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Cited by 24 (4 self)
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. We consider branch and bound methods for enclosing all unconstrained global minimizers of a nonconvex nonlinear twicecontinuously differentiable objective function. In particular, we consider bounds obtained with interval arithmetic, with the "midpoint test," but no acceleration procedures. Unless the lower bound is exact, the algorithm without acceleration procedures in general gives an undesirable cluster of boxes around each minimizer. In a previous paper, we analyzed this problem for univariate objective functions. In this paper, we generalize that analysis to multidimensional objective functions. As in the univariate case, the results show that the problem is highly related to the behavior of the objective function near the global minimizers and to the order of the corresponding interval extension. 1. Introduction and Basic Concepts Our underlying problem is: (1) find all global minimizers to f(x) subject to x 2 X; where X ae R m is a compact right parallelepiped with face...
Empirical Evaluation Of Innovations In Interval Branch And Bound Algorithms For Nonlinear Systems
 SIAM J. Sci. Comput
, 1994
"... . Interval branch and bound algorithms for finding all roots use a combination of a computational existence / uniqueness procedure and a tesselation process (generalized bisection). Such algorithms identify, with mathematical rigor, a set of boxes that contains unique roots and a second set within w ..."
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Cited by 20 (10 self)
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. Interval branch and bound algorithms for finding all roots use a combination of a computational existence / uniqueness procedure and a tesselation process (generalized bisection). Such algorithms identify, with mathematical rigor, a set of boxes that contains unique roots and a second set within which all remaining roots must lie. Though each root is contained in a box in one of the sets, the second set may have several boxes in clusters near a single root. Thus, the output is of higher quality if there are relatively more boxes in the first set. In contrast to previously implemented similar techniques, a box expansion technique in this paper, based on using an approximate root finder, fflinflation and exact set complementation, decreases the size of the second set, increases the size of the first set, and never loses roots. In addition to the expansion technique, use of secondorder extensions to eliminate small boxes that do not contain roots, and interval slopes versus interval d...
Hierarchical Arc Consistency for Disjoint Real Intervals in Constraint Logic Programming
 COMPUTATIONAL INTELLIGENCE
, 1992
"... There have been many proposals for adding sound implementations of numeric processing to Prolog. This paper describes an approach to numeric constraint processing which has been implemented in Echidna, a new constraint logic programming (CLP) language. Echidna uses consistency algorithms which can a ..."
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Cited by 19 (0 self)
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There have been many proposals for adding sound implementations of numeric processing to Prolog. This paper describes an approach to numeric constraint processing which has been implemented in Echidna, a new constraint logic programming (CLP) language. Echidna uses consistency algorithms which can actively process a wider variety of numeric constraints than most other CLP systems, including constraints containing some common nonlinear functions. A unique feature of Echidna is that it implements domains for realvalued variables with hierarchical data structures and exploits this structure using a hierarchical arc consistency algorithm specialized for numeric constraints. This gives Echidna two advantages over other systems. First, the union of disjoint intervals can be represented directly. Other approaches require trying each disjoint interval in turn during backtrack search. Second, the hierarchical structure facilitates varying the precision of constraint processing. Consequently...
Surface Intersection Using Affine Arithmetic
 In Graphics Interface
, 1996
"... We describe a variant of a domain decomposition method proposed by Gleicher and Kass for intersecting and trimming parametric surfaces. Instead of using interval arithmetic to guide the decomposition, the variant described here uses affine arithmetic, a tool recently proposed for range analysis. Aff ..."
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Cited by 18 (7 self)
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We describe a variant of a domain decomposition method proposed by Gleicher and Kass for intersecting and trimming parametric surfaces. Instead of using interval arithmetic to guide the decomposition, the variant described here uses affine arithmetic, a tool recently proposed 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. As a consequence, the quadtree domain decompositions are much smaller and the intersection algorithm runs faster. keywords: surface intersection, trimming surfaces, range analysis, interval analysis, CAGD.