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Highlevel filtering for arrangements of conic arcs
 In Proc. ESA 2002
, 2002
"... Abstract. Many computational geometry algorithms involve the construction and maintenance of planar arrangements of conic arcs. Implementing a general, robust arrangement package for conic arcs handles most practical cases of planar arrangements covered in literature. A possible approach for impleme ..."
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Cited by 33 (9 self)
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Abstract. Many computational geometry algorithms involve the construction and maintenance of planar arrangements of conic arcs. Implementing a general, robust arrangement package for conic arcs handles most practical cases of planar arrangements covered in literature. A possible approach for implementing robust geometric algorithms is to use exact algebraic number types — yet this may lead to a very slow, inefficient program. In this paper we suggest a simple technique for filtering the computations involved in the arrangement construction: when constructing an arrangement vertex, we keep track of the steps that lead to its construction and the equations we need to solve to obtain its coordinates. This construction history can be used for answering predicates very efficiently, compared to a naïve implementation with an exact number type. Furthermore, using this representation most arrangement vertices may be computed approximately at first and can be refined later on in cases of ambiguity. Since such cases are relatively rare, the resulting implementation is both efficient and robust. 1
Towards an open curved kernel
 In Proc. Annual ACM Symp. on Computational Geometry
, 2004
"... Our work goes towards answering the growing need for the robust and efficient manipulation of curved objects in numerous applications. The kernel of the cgal library provides several functionalities which are, however, mostly restricted to linear objects. We focus here on the arrangement of conic ar ..."
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Cited by 31 (14 self)
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Our work goes towards answering the growing need for the robust and efficient manipulation of curved objects in numerous applications. The kernel of the cgal library provides several functionalities which are, however, mostly restricted to linear objects. We focus here on the arrangement of conic arcs in the plane. Our first contribution is the design, implementation and testing of a kernel for computing arrangements of circular arcs. A preliminary C++ implementation exists also for arbitrary conic curves. We discuss the representation and predicates of the geometric objects. Our implementation is targeted for inclusion in the cgal library. Our second contribution concerns exact and efficient algebraic algorithms for the case of conics. They treat all inputs, including degeneracies, and they are implemented as part of the library synaps 2.1. Our tools include Sturm sequences, resultants, Descartes ’ rule, and isolating points. Thirdly, our experiments on circular arcs show that our ∗ Work partially supported by the IST Programme of the EU as a
A computational basis for conic arcs and Boolean operations on conic polygons
 IN ESA 2002, LNCS 2461
, 2002
"... We give an exact geometry kernel for conic arcs, algorithms for exact computation with lowdegree algebraic numbers, and an algorithm for computing the arrangement of conic arcs that immediately leads to a realization of regularized boolean operations on conic polygons. A conic polygon, or polygon ..."
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Cited by 29 (15 self)
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We give an exact geometry kernel for conic arcs, algorithms for exact computation with lowdegree algebraic numbers, and an algorithm for computing the arrangement of conic arcs that immediately leads to a realization of regularized boolean operations on conic polygons. A conic polygon, or polygon for short, is anything that can be obtained from linear or conic halfspaces ( = the set of points where a linear or quadratic function is nonnegative) by regularized boolean operations. The algorithm and its implementation are complete (they can handle all cases), exact (they give the mathematically correct result), and efficient (they can handle inputs with several hundred primitives).
Complete, exact, and efficient computations with cubic curves
 In Proc. 20th Annu. ACM Symp. Comput. Geom
, 2004
"... The BentleyOttmann sweepline method can be used to compute the arrangement of planar curves provided a number of geometric primitives operating on the curves are available. We discuss the mathematics of the primitives for planar algebraic curves of degree three or less and derive efficient realiza ..."
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Cited by 17 (6 self)
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The BentleyOttmann sweepline method can be used to compute the arrangement of planar curves provided a number of geometric primitives operating on the curves are available. We discuss the mathematics of the primitives for planar algebraic curves of degree three or less and derive efficient realizations. As a result, we obtain a complete, exact, and efficient algorithm for computing arrangements of cubic curves. Conics and cubic splines are special cases of cubic curves. The algorithm is complete in that it handles all possible degeneracies including singularities. It is exact in that it provides the mathematically correct result. It is efficient in that it can handle hundreds of curves with a quarter million of segments in the final arrangement.
The predicates for the Voronoi diagram of ellipses
 In Proc. 22th Annual ACM Symp. on Computational Geometry
, 2006
"... This paper examines the computation of the Voronoi diagram of a set of ellipses in the Euclidean plane. We propose the first complete algorithms, under the exact computation paradigm, for the predicates of an incremental algorithm: κ1 decides which one of 2 given ellipses is closest to a given exter ..."
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Cited by 13 (8 self)
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This paper examines the computation of the Voronoi diagram of a set of ellipses in the Euclidean plane. We propose the first complete algorithms, under the exact computation paradigm, for the predicates of an incremental algorithm: κ1 decides which one of 2 given ellipses is closest to a given exterior point; κ2 decides the position of a query ellipse relative to an external bitangent line of 2 given ellipses; κ3 decides the position of a query ellipse relative to a Voronoi circle of 3 given ellipses; κ4 determines the type of conflict between a Voronoi edge, defined by 4 given ellipses, and a query ellipse. The paper is restricted to nonintersecting ellipses, but the extension to arbitrary ones is straightforward. The ellipses are input in parametric representation or constructively. For κ1 and κ2 we derive optimal algebraic conditions, solve them exactly and provide efficient implementations in C++. For κ3 we compute a tight bound on the number of complex tritangent circles and use the parametric form of the ellipses in order to design an exact subdivisionbased algorithm, which is implemented on Maple. This approach essentially answers κ4 as well. We conclude with current work on optimizing κ3 and implementing it in C++.
Hypergeometric Functions in Exact Geometric Computation
, 2002
"... Most problems in computational geometry are algebraic. A general approach to address nonrobustness in such problems is Exact Geometric Computation (EGC). There are now general libraries that support EGC for the general programmer (e.g., Core Library, LEDA Real). But many applications require nonalg ..."
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Cited by 8 (6 self)
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Most problems in computational geometry are algebraic. A general approach to address nonrobustness in such problems is Exact Geometric Computation (EGC). There are now general libraries that support EGC for the general programmer (e.g., Core Library, LEDA Real). But many applications require nonalgebraic functions as well. In this paper, we wish to provide nonalgebraic functions in the context of other EGC capabilities. We implemented a multiprecision hypergeometric series package which can be used to evaluate the elementary functions. This can be achieved relatively easily using the Core Library which supports a guaranteed precision level of accuracy. We address several issues of efficiency in such a hypergeometric package: automatic error analysis, argument reduction, preprocessing of hypergeometric parameters, and precomputed constants. Some preliminary experimental results are reported.
On the Topology of Planar Algebraic Curves
"... We introduce a method to compute the topology of planar algebraic curves. The curve may not be in generic position and may have vertical asymptotes. The algebraic tools are rational univariate representation for zerodimentional ideals and multiplicities in these ideals. Experiments show the e cienc ..."
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Cited by 7 (1 self)
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We introduce a method to compute the topology of planar algebraic curves. The curve may not be in generic position and may have vertical asymptotes. The algebraic tools are rational univariate representation for zerodimentional ideals and multiplicities in these ideals. Experiments show the e ciency of our algorithm. 1
Planar embeddings of graphs with specified edge lengths
, 2007
"... We consider the problem of finding a planar straightline embedding of a graph with a prescribed Euclidean length on every edge. There has been substantial previous work on the problem without the planarity restrictions, which has close connections to rigidity theory, and where it is easy to see tha ..."
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Cited by 6 (2 self)
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We consider the problem of finding a planar straightline embedding of a graph with a prescribed Euclidean length on every edge. There has been substantial previous work on the problem without the planarity restrictions, which has close connections to rigidity theory, and where it is easy to see that the problem is NPhard. In contrast, we show that the problem is tractable—indeed, solvable in linear time on a real RAM—for straightline embeddings of planar 3connected triangulations, even if the outer face is not a triangle. This result is essentially tight: the problem becomes NPhard if we consider instead straightline embeddings of planar 3connected infinitesimally rigid graphs with unit edge lengths, a natural relaxation of triangulations in this context.
Toward the Integration of Numerical Computations into the OMSCS Framework
"... Abstract. Computer algebra systems and automated theorem provers, which have complementary abilities, can be integrated to form an Open Mechanized Symbolic Computation System (OMSCS). This framework could be extended to integrate numerical computation systems. This paper aims at showing what problem ..."
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Cited by 4 (3 self)
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Abstract. Computer algebra systems and automated theorem provers, which have complementary abilities, can be integrated to form an Open Mechanized Symbolic Computation System (OMSCS). This framework could be extended to integrate numerical computation systems. This paper aims at showing what problems can occur when dealing with numerical computations and what can be done to solve them or at least to provide a clear meaning of a numerical result; it constitutes a step toward this integration. 1