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111
Static Analysis Yields Efficient Exact Integer Arithmetic for Computational Geometry
 ACM Trans. Graph
, 1996
"... Geometric algorithms are usually described assuming that arithmetic operations are performed exactly on real numbers. A program implemented using a naive substitution of floatingpoint arithmetic for real arithmetic can fail, since geometric primitives depend upon signevaluation and may not be re ..."
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Cited by 58 (4 self)
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Geometric algorithms are usually described assuming that arithmetic operations are performed exactly on real numbers. A program implemented using a naive substitution of floatingpoint arithmetic for real arithmetic can fail, since geometric primitives depend upon signevaluation and may not be reliable if evaluated approximately. Geometric primitives are reliable if evaluated exactly with integer arithmetic, but this degrades performance since software extendedprecision arithmetic is required. We describe staticanalysis techniques that reduce the performance cost of exact integer arithmetic used to implement geometric algorithms. We have used the techniques for a number of examples, including linesegment intersection in two dimensions, Delaunay triangulations, and a threedimensional boundarybased polyhedral modeller. In general, the techniques are appropriate for algorithms that use primitives of relatively low algebraic total degree, e.g., those involving flat objects (...
Implicitization using Moving Curves and Surfaces
, 1995
"... This paper presents a radically new approach to the century old problem of computing the implicit equation of a parametric surface. For surfaces without base points, the new method expresses the implicit equation in a determinant which is one fourth the size of the conventional expression based on D ..."
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Cited by 54 (3 self)
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This paper presents a radically new approach to the century old problem of computing the implicit equation of a parametric surface. For surfaces without base points, the new method expresses the implicit equation in a determinant which is one fourth the size of the conventional expression based on Dixon's resultant. If base points do exist, previous implicitization methods either fail or become much more complicated, while the new method actually simplifies.
Using Generic Programming for Designing a Data Structure for Polyhedral Surfaces
 Comput. Geom. Theory Appl
, 1999
"... Appeared in Computational Geometry  Theory and Applications 13, 1999, 6590. Software design solutions are presented for combinatorial data structures, such as polyhedral surfaces and planar maps, tailored for program libraries in computational geometry. Design issues considered are flexibility, ..."
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Cited by 47 (6 self)
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Appeared in Computational Geometry  Theory and Applications 13, 1999, 6590. Software design solutions are presented for combinatorial data structures, such as polyhedral surfaces and planar maps, tailored for program libraries in computational geometry. Design issues considered are flexibility, time and space efficiency, and easeofuse. We focus on topological aspects of polyhedral surfaces and evaluate edgebased representations with respect to our design goals. A design for polyhedral surfaces in a halfedge data structure is developed following the generic programming paradigm known from the Standard Template Library STL for C++. Connections are shown to planar maps and facebased structures. Key words: Library design; Generic programming; Combinatorial data structure; Polyhedral surface; Halfedge data structure 1 Introduction Combinatorial structures, such as planar maps, are fundamental in computational geometry. In order to be useful in practice, a solid library for compu...
Approximate Boolean Operations on Freeform Solids
, 2001
"... In this paper we describe a method for computing approximate results of boolean operations (union, intersection, difference) applied to freeform solids bounded by multiresolution subdivision surfaces. ..."
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Cited by 36 (7 self)
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In this paper we describe a method for computing approximate results of boolean operations (union, intersection, difference) applied to freeform solids bounded by multiresolution subdivision surfaces.
Matchmaker: Manifold BReps for nonmanifold rsets
 Proceedings of the ACM Symposium on Solid Modeling
, 1999
"... Many solid modeling construction techniques produce nonmanifold rsets (solids). With each nonmanifold model N we can associate a family of manifold solid models that are infinitely close to N in the geometric sense. For polyhedral solids, each nonmanifold edge of N with 2k incident faces will be ..."
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Cited by 36 (17 self)
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Many solid modeling construction techniques produce nonmanifold rsets (solids). With each nonmanifold model N we can associate a family of manifold solid models that are infinitely close to N in the geometric sense. For polyhedral solids, each nonmanifold edge of N with 2k incident faces will be replicated k times in any manifold model M of that family. Furthermore, some nonmanifold vertices of N must also be replicated in M, possibly several times. M can be obtained by defining, in N, a single adjacent face TA(E,F) for each pair (E,F) that combines an edge E and an incident face F. The adjacency relation satisfies TA(E,TA(E,F))=F. The choice of the map A defines which vertices of N must be replicated in M and how many times. The resulting manifold representation of a nonmanifold solid may be encoded using simpler and more compact datastructures, especially for triangulated model, and leads to simpler and more efficient algorithms, when it is used instead of a nonmanifold repre...
Interactive Boolean Operations for Conceptual Design of 3D Solids
, 1997
"... Interactive modeling of 3D solids is an important and difficult problem in computer graphics. The Constructive Solid Geometry (CSG) modeling scheme is highly attractive for interactive design, due to its support for hierarchical modeling and Boolean operations. Unfortunately, current algorithms for ..."
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Cited by 36 (1 self)
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Interactive modeling of 3D solids is an important and difficult problem in computer graphics. The Constructive Solid Geometry (CSG) modeling scheme is highly attractive for interactive design, due to its support for hierarchical modeling and Boolean operations. Unfortunately, current algorithms for interactive display of CSG models require expensive specialpurpose hardware that is not easily available. In this paper we present a method for interactive display of CSG models using standard, widely available graphics hardware. The method enables the user to interactively modify the affine transformations associated with CSG subobjects. The application we focus upon is that of conceptual design, a stage in the design process in which rapid, interactive visualization of the model and highlevel design operations are of crucial importance, while the objects are relatively simple. The method converts the CSG graph to a novel Convex Differences Aggregate(CDA) representation. The CDA utili...
Converting bases with the Gröbner walk
 Journal of Symbolic Computation
, 1997
"... We present an algorithm which converts a given Gröbner basis of a polynomial ideal I to a Gröbner basis of I with respect to another term order. The conversion is done in several steps following a path in the Gröbner fan of I. Each conversion step is based on the computation of a Gröbner basis of a ..."
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Cited by 32 (1 self)
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We present an algorithm which converts a given Gröbner basis of a polynomial ideal I to a Gröbner basis of I with respect to another term order. The conversion is done in several steps following a path in the Gröbner fan of I. Each conversion step is based on the computation of a Gröbner basis of a toric degeneration of I. c ○ 1997 Academic Press Limited 1.
Designing a Data Structure for Polyhedral Surfaces
 In Proc. 14th Annu. ACM Sympos. Comput. Geom
, 1998
"... Design solutions for a program library are presented for combinatorial data structures in computational geometry, such as planar maps and polyhedral surfaces. Design issues considered are genericity, flexibility, time and space efficiency, and easeofuse. We focus on topological aspects of polyhedr ..."
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Cited by 31 (2 self)
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Design solutions for a program library are presented for combinatorial data structures in computational geometry, such as planar maps and polyhedral surfaces. Design issues considered are genericity, flexibility, time and space efficiency, and easeofuse. We focus on topological aspects of polyhedral surfaces. Edgebased representations for polyhedrons are evaluated with respect to the design goals. A design for polyhedral surfaces in a halfedge data structure is developed following the generic programming paradigm known from the Standard Template Library STL for C++. Connections are shown to planar maps and facebased structures managing holes in facets. 1 Introduction Combinatorial structures, such as planar maps, are fundamental in computational geometry. In order to use computational geometry in practice, a solid library must provide generic and flexible solutions as one of its fundamental cornerstones. Other design criteria are time and space efficiency. Easeofuse is necessar...
Preventing selfintersection under freeform deformation
 IEEE Transactions on Visualization and Computer Graphics
, 2001
"... AbstractÐFreeForm Deformation (FFD) is a versatile and efficient modeling technique which transforms an object by warping the surrounding space. The conventional userinterface is a lattice of movable control points but this tends to be cumbersome and counterintuitive. Directly Manipulated FreeFor ..."
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Cited by 27 (3 self)
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AbstractÐFreeForm Deformation (FFD) is a versatile and efficient modeling technique which transforms an object by warping the surrounding space. The conventional userinterface is a lattice of movable control points but this tends to be cumbersome and counterintuitive. Directly Manipulated FreeForm Deformation (DMFFD) allows the user to drag object points directly and has proven useful in an interactive sculpting context. A serious shortcoming of both FFD and DMFFD is that some deformations cause selfintersection of the object. This is unrealistic and compromises the object's validity and suitability for later use. An inbuilt selfintersection test is thus required for FFD and its extensions to be truly robust. In this paper, we present the following novel results: a set of theoretical conditions for preventing selfintersection by ensuring the injectivity (onetoone mapping) of FFD, an exact (necessary and sufficient) injectivity test which is accurate but computationally costly, an efficient but approximate injectivity test which is a sufficient condition only, and a new form of DMFFD which acts by composing many small injective deformations. The latter expands the range of possible deformations without sacrificing the speed of the approximate test. Index TermsÐFreeform deformation, direct manipulation, selfintersection, space homeomorphism. 1