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Advanced programming techniques applied to Cgal’s arrangement package
 Computational Geometry: Theory and Applications
, 2005
"... Arrangements of planar curves are fundamental structures in computational geometry. Recently, the arrangement package of Cgal, the Computational Geometry Algorithms Library, has been redesigned and reimplemented exploiting several advanced programming techniques. The resulting software package, whi ..."
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Cited by 31 (15 self)
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Arrangements of planar curves are fundamental structures in computational geometry. Recently, the arrangement package of Cgal, the Computational Geometry Algorithms Library, has been redesigned and reimplemented exploiting several advanced programming techniques. The resulting software package, which constructs and maintains planar arrangements, is easier to use, to extend, and to adapt to a variety of applications. It is more efficient space and timewise, and more robust. The implementation is complete in the sense that it handles degenerate input, and it produces exact results. In this paper we describe how various programming techniques were used to accomplish specific tasks within the context of computational geometry in general and Arrangements in particular. These tasks are exemplified by several applications, whose robust implementation is based on the arrangement package. Together with a set of benchmarks they assured the successful application of the adverted programming techniques. 1
Exacus: Efficient and exact algorithms for curves and surfaces
 IN ESA, VOLUME 1669 OF LNCS
, 2005
"... We present the first release of the EXACUS C++ libraries. We aim for systematic support of nonlinear geometry in software libraries. Our goals are efficiency, correctness, completeness, clarity of the design, modularity, flexibility, and ease of use. We present the generic design and structure of ..."
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Cited by 30 (12 self)
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We present the first release of the EXACUS C++ libraries. We aim for systematic support of nonlinear geometry in software libraries. Our goals are efficiency, correctness, completeness, clarity of the design, modularity, flexibility, and ease of use. We present the generic design and structure of the libraries, which currently compute arrangements of curves and curve segments of low algebraic degree, and boolean operations on polygons bounded by such segments.
Real Algebraic Numbers: Complexity Analysis and Experimentation
 RELIABLE IMPLEMENTATIONS OF REAL NUMBER ALGORITHMS: THEORY AND PRACTICE, LNCS (TO APPEAR
, 2006
"... We present algorithmic, complexity and implementation results concerning real root isolation of a polynomial of degree d, with integer coefficients of bit size ≤ τ, using Sturm (Habicht) sequences and the Bernstein subdivision solver. In particular, we unify and simplify the analysis of both metho ..."
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Cited by 30 (17 self)
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We present algorithmic, complexity and implementation results concerning real root isolation of a polynomial of degree d, with integer coefficients of bit size ≤ τ, using Sturm (Habicht) sequences and the Bernstein subdivision solver. In particular, we unify and simplify the analysis of both methods and we give an asymptotic complexity bound of eOB(d 4 τ 2). This matches the best known bounds for binary subdivision solvers. Moreover, we generalize this to cover the non squarefree polynomials and show that within the same complexity we can also compute the multiplicities of the roots. We also consider algorithms for sign evaluation, comparison of real algebraic numbers and simultaneous inequalities, and we improve the known bounds at least by a factor of d. Finally, we present our C++ implementation in synaps and some preliminary experiments on various data sets.
An exact, complete and efficient implementation for computing planar maps of quadric intersection curves
 In Proc. 21th Annu. Sympos. Comput. Geom
, 2005
"... Ú�ÖØ��×�ÒØ��×�Ö�Ô��Ö�Ø��×�Ò�ÙÐ�Ö�Ò�Ü�ÜØÖ�Ñ�ÔÓ�ÒØ × Ó�Ø��ÙÖÚ�×�×Û�ÐÐ�×�ÐÐ�ÒØ�Ö×�Ø�ÓÒÔÓ�ÒØ×Ó�Ô��Ö×Ó � ÙÖÚ�×Ô�Ô����ÒÖÙÒÒ�Ò�ÓÒØ��×ÙÖ���Ó�ÔÌ� � ÙÖÚ�×ÌÛÓÚ�ÖØ��×�Ö�ÓÒÒ�Ø���Ý�Ò������Ø��ÙÒ��Ö ��Ò�Ð��ÐÐ��Ò�Ó��ÒÔÙØ×�ÒÐÙ��Ò��ÐÐ����Ò�Ö�Ø�ÓÒ�×Û��Ö � �Ò��� ℄ ÇÙÖÛÓÖ��×��×��ÓÒ�Ò��ÜØ�Ò�×����×��Ú�ÐÓÔ���Ò� ℄ ÐÝ�Ò�ÔÓ�ÒØ ..."
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Cited by 26 (12 self)
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Ú�ÖØ��×�ÒØ��×�Ö�Ô��Ö�Ø��×�Ò�ÙÐ�Ö�Ò�Ü�ÜØÖ�Ñ�ÔÓ�ÒØ × Ó�Ø��ÙÖÚ�×�×Û�ÐÐ�×�ÐÐ�ÒØ�Ö×�Ø�ÓÒÔÓ�ÒØ×Ó�Ô��Ö×Ó � ÙÖÚ�×Ô�Ô����ÒÖÙÒÒ�Ò�ÓÒØ��×ÙÖ���Ó�ÔÌ� � ÙÖÚ�×ÌÛÓÚ�ÖØ��×�Ö�ÓÒÒ�Ø���Ý�Ò������Ø��ÙÒ��Ö ��Ò�Ð��ÐÐ��Ò�Ó��ÒÔÙØ×�ÒÐÙ��Ò��ÐÐ����Ò�Ö�Ø�ÓÒ�×Û��Ö � �Ò��� ℄ ÇÙÖÛÓÖ��×��×��ÓÒ�Ò��ÜØ�Ò�×����×��Ú�ÐÓÔ���Ò� ℄ ÐÝ�Ò�ÔÓ�ÒØ×�Ö�ÓÒÒ�Ø���Ý��Ö�Ò�Ó�ÓÒ�Ó�Ø��ÙÖÚ�× �ÒØ�Ö×�Ø�ÓÒÙÖÚ�×��Ú�×�Ò�ÙÐ�Ö�Ø��×ÓÖÔ��Ö×Ó�ÙÖÚ�×�Ò Ø�Ö×�ØÛ�Ø�����ÑÙÐØ�ÔÐ��ØÝÁØ�×�Ü�Ø�ÒØ��Ø�Ø�ÐÛ�Ý × ÓÑÔÙØ�×Ø��Ñ�Ø��Ñ�Ø��ÐÓÖÖ�ØÖ�×ÙÐØÁØ�×�Æ��ÒØ ÇÙÖ�ÑÔÐ�Ñ�ÒØ�Ø�ÓÒ�×ÓÑÔÐ�Ø��ÒØ��×�Ò×�Ø��Ø�Ø�Ò Ñ��×ÙÖ���ÒÖÙÒÒ�Ò�Ø�Ñ�× Ç���ØÅÓ��Ð�Ò����ÓÑ�ØÖ��Ð�ÓÖ�Ø�Ñ×����ÆÙÑ�Ö��Ð �Ò�ÐÝ×�×℄�ÊÓÓØ×Ó�ÆÓÒÐ�Ò��Ö�ÕÙ�Ø�ÓÒ×�Ñ�Ø�Ó�×�ÓÖÔÓÐÝ Á���ÓÑÔÙØ�Ö�Ö�Ô��×℄��ÓÑÔÙØ�Ø�ÓÒ�Ð��ÓÑ�ØÖÝ�Ò � £È�ÖØ��ÐÐÝ×ÙÔÔÓÖØ���ÝØ��ÁËÌÈÖÓ�Ö�ÑÑ�Ó�Ø���ÙÖÓ Ô��ÒÍÒ�ÓÒ�×�Ë��Ö��Ó×ØÊÌ���ÌÇÔ�ÒÈÖÓ��ØÙÒ ��Ö�ÓÒØÖ�ØÆÓÁËÌ Categories Ø�Ø�ÓÒ�Ð��ÓÑ�ØÖÝ�ÓÖ�ÙÖÚ�×�Ò�ËÙÖ��� × ������ß�«�Ø�Ú��ÓÑÔÙ and Subject Descriptors ��Å ÁØ�×ÔÓ×Ø����Ö��ÝÔ�ÖÑ�××�ÓÒÓ���Å�ÓÖÝÓÙÖÔ�Ö ×ÓÒ�ÐÙ×�ÆÓØ�ÓÖÖ���×ØÖ��ÙØ�ÓÒÌ����¬Ò�Ø�Ú�Ú�Ö ×�ÓÒÛ�×ÔÙ�Ð�×����ÒØ��ÈÖÓ����Ò�×Ó�Ø� � �Ì��×�×Ø���ÙØ�ÓÖ×Ú�Ö×�ÓÒÓ�Ø��ÛÓÖ� �ØØÔ��Ó��ÑÓÖ � ÒÙ�ÐËÝÑÔÓ×�ÙÑÓÒ�ÓÑÔÙØ�Ø�ÓÒ�Ð��ÓÑ�ØÖÝË� � ���� � � � ×Ø�Ò �
An adaptable and extensible geometry kernel
 In Proc. Workshop on Algorithm Engineering
, 2001
"... ii ..."
Comparison Of FourthDegree Algebraic Numbers And Applications To Geometric Predicates
, 2000
"... We present algorithms for the exact comparison of the real roots of two polynomials of degree 4. The algorithm precomputes Sturm sequences and isolating intervals for the representation of the roots and, additionally, uses various invariants in order to minimize the computational e#ort. In most c ..."
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Cited by 12 (5 self)
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We present algorithms for the exact comparison of the real roots of two polynomials of degree 4. The algorithm precomputes Sturm sequences and isolating intervals for the representation of the roots and, additionally, uses various invariants in order to minimize the computational e#ort. In most cases, the algorithm is optimal with respect to the algebraic degree of the tested quantities in the input coe#cients. Our treatment is complete, in the sense that we handle all special cases, including when one of the polynomials has degree smaller than 4. Our algorithms have been implemented, and some preliminary experimental results are presented in order to show their e#ciency when compared to the CORE library. We apply
Real solving of bivariate polynomial systems
 Proc. Computer Algebra in Scientific Computing (CASC), LNCS
, 2005
"... Abstract. We propose exact, complete and efficient methods for 2 problems: First, the real solving of systems of two bivariate rational polynomials of arbitrary degree. This means isolating all common real solutions in rational rectangles and calculating the respective multiplicities. Second, the co ..."
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Cited by 11 (8 self)
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Abstract. We propose exact, complete and efficient methods for 2 problems: First, the real solving of systems of two bivariate rational polynomials of arbitrary degree. This means isolating all common real solutions in rational rectangles and calculating the respective multiplicities. Second, the computation of the sign of bivariate polynomials evaluated at two algebraic numbers of arbitrary degree. Our main motivation comes from nonlinear computational geometry and computeraided design, where bivariate polynomials lie at the inner loop of many algorithms. The methods employed are based on SturmHabicht sequences, univariate resultants and rational univariate representation. We have implemented them very carefully, using advanced objectoriented programming techniques, so as to achieve high practical performance. The algorithms are integrated in the publicdomain C++ software library synaps, and their efficiency is illustrated by 9 experiments against existing implementations. Our code is faster in most cases; sometimes it is even faster than numerical approaches. 1
An Exact and Efficient Approach for Computing a Cell in an Arrangement of Quadrics
, 2006
"... We present an approach for the exact and efficient computation of a cell in an arrangement of quadric surfaces. All calculations are based on exact rational algebraic methods and provide the correct mathematical results in all, even degenerate, cases. By projection, the spatial problem is reduced to ..."
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Cited by 7 (1 self)
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We present an approach for the exact and efficient computation of a cell in an arrangement of quadric surfaces. All calculations are based on exact rational algebraic methods and provide the correct mathematical results in all, even degenerate, cases. By projection, the spatial problem is reduced to the one of computing planar arrangements of algebraic curves. We succeed in locating all event points in these arrangements, including tangential intersections and singular points. By introducing an additional curve, which we call the Jacobi curve, we are able to find nonsingular tangential intersections. We show that the coordinates of the singular points in our special projected planar arrangements are roots of quadratic polynomials. The coefficients of these polynomials are usually rational and contain at most a single square root. A prototypical implementation indicates that our approach leads to good performance in practice.
COMPUTATIONAL AND STRUCTURAL ADVANTAGES OF CIRCULAR BOUNDARY REPRESENTATION
, 2009
"... Boundary approximation of planar shapes by circular arcs has quantitative and qualitative advantages compared to using straightline segments. We demonstrate this by way of three basic and frequent computations on shapes – convex hull, decomposition, and medial axis. In particular, we propose a nov ..."
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Cited by 5 (3 self)
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Boundary approximation of planar shapes by circular arcs has quantitative and qualitative advantages compared to using straightline segments. We demonstrate this by way of three basic and frequent computations on shapes – convex hull, decomposition, and medial axis. In particular, we propose a novel medial axis algorithm that beats existing methods in simplicity and practicality, and at the same time guarantees convergence to the medial axis of the original shape.
Exact, Efficient, and Complete Arrangement Computation for Cubic Curves
, 2006
"... The BentleyOttmann sweepline method can compute the arrangement of planar curves, provided a number of geometric primitives operating on the curves are available. We discuss the reduction of the primitives to the analysis of curves and curve pairs, and describe efficient realizations of these anal ..."
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Cited by 3 (0 self)
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The BentleyOttmann sweepline method can compute the arrangement of planar curves, provided a number of geometric primitives operating on the curves are available. We discuss the reduction of the primitives to the analysis of curves and curve pairs, and describe efficient realizations of these analyses for planar algebraic curves of degree three or less. We obtain a complete, exact, and efficient algorithm for computing arrangements of cubic curves. Special cases of cubic curves are conics as well as implicitized cubic splines and Bézier curves. The algorithm is complete in that it handles all possible degeneracies such as tangential intersections and 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 algorithm has been implemented in C++ as an Exacus library called CubiX.