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16
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...
Interval Methods for MultiPoint Collisions between TimeDependent Curved Surfaces
 Computer Graphics
, 1993
"... We present an efficient and robust algorithm for finding points of collision between timedependent parametric and implicit surfaces. The algorithm detects simultaneous collisions at multiple points of contact. When the regions of contact form curves or surfaces, it returns a finite set of points un ..."
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Cited by 63 (0 self)
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We present an efficient and robust algorithm for finding points of collision between timedependent parametric and implicit surfaces. The algorithm detects simultaneous collisions at multiple points of contact. When the regions of contact form curves or surfaces, it returns a finite set of points uniformly distributed over each contact region. Collisions can be computed for a very general class of surfaces: those for which inclusion functions can be constructed. Included in this set are the familiar kinds of surfaces and time behaviors encountered in computer graphics. We use a new interval approach for constrained minimization to detect collisions, and a tangency condition to reduce the dimensionality of the search space. These approaches make interval methods practical for multipoint collisions between complex surfaces. An interval Newton method based on the solution of the interval linear equation is used to speed convergence to the collision time and location. This method is mor...
Computation and Application of Taylor Polynomials with Interval Remainder Bounds
 Reliable Computing
, 1998
"... . The expansion of complicated functions of many variables in Taylor polynomials is an important problem for many applications, and in practice can be performed rather conveniently (even to high orders) using polynomial algebras. An important application of these methods is the field of beam physics ..."
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Cited by 34 (2 self)
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. The expansion of complicated functions of many variables in Taylor polynomials is an important problem for many applications, and in practice can be performed rather conveniently (even to high orders) using polynomial algebras. An important application of these methods is the field of beam physics, where often expansions in about six variables to orders between five and ten are used. However, often it is necessary to also know bounds for the remainder term of the Taylor formula if the arguments lie within certain intervals. In principle such bounds can be obtained by interval bounding of the (n+1)st derivative, which in turn can be obtained with polynomial algebra; but in practice the method is rather inefficient and susceptible to blowup because of the need of repeated interval evaluations of the derivative. Here we present a new method that allows the computation of sharp remainder intervals in parallel with the accumulation derivatives up to order n. The method is useful for a...
A Heuristic Rejection Criterion in Interval Global Optimization Algorithms
, 1999
"... This paper investigates the properties of the inclusion functions on subintervals while a BranchandBound algorithm is solving global optimization problems. It has been found that the relative place of the global minimum value within the inclusion interval of the inclusion function of the objective ..."
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Cited by 11 (7 self)
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This paper investigates the properties of the inclusion functions on subintervals while a BranchandBound algorithm is solving global optimization problems. It has been found that the relative place of the global minimum value within the inclusion interval of the inclusion function of the objective function at the actual interval mostly indicates whether the given interval is close to a minimizer point. This information is used in a heuristic interval rejection rule that can save a big amount of computation. Illustrative examples are discussed and a numerical study completes the investigation. AMS subject classication: 65K, 90C. Key words: Global optimization, BranchandBound Algorithm, Inclusion Function. 1
Optimization and the Miranda approach in detecting horseshoetype chaos by computer
 Int. J. Bifurcation and Chaos
, 2007
"... We report on experiences with an adaptive subdivision method supported by interval arithmetic that enables us to prove subset relations of the form T (W) ⊂ U and thus to check certain sufficient conditions for chaotic behaviour of dynamical systems in a rigorous way. Our proof of the underlying abs ..."
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Cited by 10 (6 self)
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We report on experiences with an adaptive subdivision method supported by interval arithmetic that enables us to prove subset relations of the form T (W) ⊂ U and thus to check certain sufficient conditions for chaotic behaviour of dynamical systems in a rigorous way. Our proof of the underlying abstract theorem avoids of referring to any results of applied algebraic topology and relies only on the Brouwer fixed point theorem. The second novelty is that the process of gaining the subset relations to be checked is, to a large extent, also automatized. The promising subset relations come from solving a constrained optimization problem via the penalty function approach. Abstract results and computational methods are demonstrated by finding embedded copies of the standard horseshoe dynamics in iterates of the classical Hénon mapping.
LIA InC++: A Local Interval Arithmetic Library for Discontinuous Intervals
, 1995
"... This paper documents LIA InC++ library for local interval arithmetic in C++. The main innovation of the library is the idea of extending traditional interval arithmetic with "complement" intervals and discontinuous intervals. By these extensions it is possible to evaluate not only ranges of possible ..."
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Cited by 8 (5 self)
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This paper documents LIA InC++ library for local interval arithmetic in C++. The main innovation of the library is the idea of extending traditional interval arithmetic with "complement" intervals and discontinuous intervals. By these extensions it is possible to evaluate not only ranges of possible values (i.e., intervals) but ranges of impossible values as well. LIA InC++ contains classes for interval types and overloaded definitions for primitive interval arithmetic operators and functions. Open ended intervals can be used in addition to the traditional closed ones. Intervals of infinite width (e.g., (,2], (,),...) are accepted as inputs and are used for managing problems of overflowing values during function evaluation. The library uses double precision machine arithmetic with optional outward rounding, makes use of interval properties such as scalarity and symmetry, and uses some bitlevel manipulations for efficient computation. LIA library is the most fundamental one in our In...
InC++ Library Family for Interval Computations
 INTERNATIONAL JOURNAL OF RELIABLE COMPUTING. SUPPLEMENT TO THE INTERNATIONAL WORKSHOP ON APPLICATIONS OF INTERVAL COMPUTATIONS
, 1995
"... This paper presents a series of C++ libraries for interval function evaluation and constraint satisfaction. Classical interval arithmetic (IA) (Moore, 1966) is extended by open ended intervals, the notion of infinity and by "complement" and discontinuous intervals. Both algebraic and numerical IA te ..."
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Cited by 8 (4 self)
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This paper presents a series of C++ libraries for interval function evaluation and constraint satisfaction. Classical interval arithmetic (IA) (Moore, 1966) is extended by open ended intervals, the notion of infinity and by "complement" and discontinuous intervals. Both algebraic and numerical IA techniques are combined for obtaining the actual range of interval functions efficiently and for determining better than local solutions for interval constraint satisfaction problems. Our practical goal is a set of portable C++ libraries that can be used in applications without deep understanding of interval analysis.
ICE InC++: A Library for Interval Constraint Equations
, 1994
"... This paper presents a general purpose C++ library ICE InC++ for solving interval constraint satisfaction problems by local and some global tolerance propagation schemes. The library makes use of algebraic techniques for solving and simplifying solutions functions corresponding to individual constrai ..."
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Cited by 5 (4 self)
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This paper presents a general purpose C++ library ICE InC++ for solving interval constraint satisfaction problems by local and some global tolerance propagation schemes. The library makes use of algebraic techniques for solving and simplifying solutions functions corresponding to individual constraint equations, and applies numerical interval branchandbound algorithms for evaluating interval functions globally. The practical goal of the work is to provide C++ programmers with an interval constraint satisfaction library that can be used without deeper understanding of interval analysis. ii PREFACE This work is part of a project on applications of constraint propagation in designing and planning (19921994) funded mainly by Technology Development Centre of Finland (TEKES) and carried out by VTT Information Technology, Information Systems (formerly Laboratory for Information Processing). Trema Inc. and ViSolutions Inc. have acted as industrial partners in the project. Members of the s...
GIA InC++: A Global Interval Arithmetic Library for Discontinuous Intervals
"... This document discusses theoretical and practical problems of evaluating interval arithmetic (IA) functions globally, and presents a C++ library called GIA InC++ (Global Interval Arithmetic in C++) for the task. In this implementation both algebraic and numerical IA techniques are employed. Our goal ..."
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Cited by 4 (4 self)
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This document discusses theoretical and practical problems of evaluating interval arithmetic (IA) functions globally, and presents a C++ library called GIA InC++ (Global Interval Arithmetic in C++) for the task. In this implementation both algebraic and numerical IA techniques are employed. Our goal has been to provide the user with an easy to use, portable evaluator that can be applied in practical applications without deep understanding of interval analysis. The paper can be used as a user's guide to the system. ii PREFACE The work is part of a project on applications of constraint propagation in designing and planning (19921994) funded mainly by Technology Development Centre of Finland (TEKES) and carried out by VTT Information Technology, Information Systems (formerly Laboratory for Information Processing). Trema Inc. and ViSolutions Inc. have acted as industrial partners in the project. Members of the steering committee were Dr. Tech. Juha Hynynen (chairman, ViSolutions Inc.), ...
Interval Constraint Programming in C++
"... This paper discusses, how to extend C++ language with classes for interval function evaluation and constraint satisfaction. Three portable general purpose libraries for the tasks are being implemented: INT++, IAF++ and INC++. The main innovation of these systems is combination of algebraic and numer ..."
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Cited by 4 (2 self)
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This paper discusses, how to extend C++ language with classes for interval function evaluation and constraint satisfaction. Three portable general purpose libraries for the tasks are being implemented: INT++, IAF++ and INC++. The main innovation of these systems is combination of algebraic and numerical IA techniques with constraint satisfaction techniques for evaluating interval functions globally and for obtaining better than local solutions for interval constraint satisfaction problems. Our practical goal is to provide the main stream programmer with easy to use, portable C++ libraries that can be used in applications without deep understanding of interval analysis.