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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...
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 69 (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 46 (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...
Contour generators of evolving implicit surfaces
 In SM ’03: Proceedings of the eighth ACM symposium on Solid modeling and applications
, 2003
"... The contour generator is an important visibility feature of a smooth object seen under parallel projection. It is the curve on the surface which seperates frontfacing regions from backfacing regions. The apparent contour is the projection of the contour generator onto a plane perpendicular to the ..."
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Cited by 18 (3 self)
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The contour generator is an important visibility feature of a smooth object seen under parallel projection. It is the curve on the surface which seperates frontfacing regions from backfacing regions. The apparent contour is the projection of the contour generator onto a plane perpendicular to the view direction. Both curves play an important role in computer graphics. Our goal is to obtain fast and robust algorithms that compute the contour generator with a guarantee of topological correctness. To this end, we first study the singularities of the contour generator and the apparent contour, for generic views, and for generic timedependent projections, e.g. when the surface is rotated or deformed. The singularities indicate when components of the contour generator merge or split as time evolves. We present an algorithm to compute an initial contour generator, using a dynamic step size. An interval test guarantees the topological correctness. This initial contour generator can then be maintained under a timedependent projection by examining its singularities.
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 12 (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.
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
InC++: A local interval arithmetic library
 Research Report, (VTT, Technical Research Centre of Finland, Information Technology
, 1994
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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 numer ..."
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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.
InC++: A library for interval constraint equations
 Research Report, (VTT, Technical Research Centre of Finland, Information Technology
, 1994
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GIA InC++: A global interval arithmetic library
 Research Report, (VTT, Technical Research Centre of Finland, Information Technology
, 1994
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