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Computing the Minimum Diameter for Moving Points: An Exact Implementation Using Parametric Search
, 1999
"... p(t) and q(t) at time t by d(p(t); q(t)). We dene the diameter D(t), at time t, of a set S of moving points as the largest Euclidean distance among all pairs of points at time t. To be more precise D(t) = maxfd(p(t); q(t)) : p; q 2 Sg: Problem 1.1 (The diameter problem for moving points) We are g ..."
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p(t) and q(t) at time t by d(p(t); q(t)). We dene the diameter D(t), at time t, of a set S of moving points as the largest Euclidean distance among all pairs of points at time t. To be more precise D(t) = maxfd(p(t); q(t)) : p; q 2 Sg: Problem 1.1 (The diameter problem for moving points) We are given a set of n points in the plane that are moving at constant but possibly dierent velocities. We want to compute the time t at which the diameter D(t ) is minimum. 2 Moving Poin
Online zone construction in arrangements of lines in the plane
 Proc. of the 3rd Workshop of Algorithm Engineering
, 1999
"... Given a finite set L of lines in the plane we wish to compute the zone of an additional curve
in the arrangement A(L), namely the set of faces of the planar subdivision induced by the lines in L that are crossed by
, where
is not given in advance but rather provided online portion by portion. ..."
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Given a finite set L of lines in the plane we wish to compute the zone of an additional curve
in the arrangement A(L), namely the set of faces of the planar subdivision induced by the lines in L that are crossed by
, where
is not given in advance but rather provided online portion by portion. This problem is motivated by the computation of the area bisectors of a polygonal set in the plane. We present four algorithms which solve this problem efficiently and exactly (giving precise results even on degenerate input). Our main algorithm is a novel approach based on the binary plane partition technique. We implemented all four algorithms. We present implementation details, comparison of performance, and a discussion of the advantages and shortcomings of each of the proposed algorithms.
Applications of the Generic Programming Paradigm in the Design of CGAL
"... . We report on the use of the generic programming paradigm in the computational geometry algorithms library cgal. The parameterization of the geometric algorithms in cgal enhances exibility and adaptability and opens an easy way for abolishing precision and robustness problems by exact but never ..."
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. We report on the use of the generic programming paradigm in the computational geometry algorithms library cgal. The parameterization of the geometric algorithms in cgal enhances exibility and adaptability and opens an easy way for abolishing precision and robustness problems by exact but nevertheless ecient computation. Furthermore we discuss circulators, which are an extension of the iterator concept to circular structures. Such structures arise frequently in geometric computing. 1 Introduction cgal is a C++ library of geometric algorithms and data structures. It is developed by several sites in Europe and Israel. The goal is to enhance the technology transfer of the algorithmic knowledge developed in the eld of computational geometry to applications in industry and academia. Computational geometry is the subarea of algorithm design that deals with the design and analysis of algorithms for geometric problems involving objects like points, lines, polygons, and polyhedra....
Combinatorial Curve Reconstruction and the Efficient Exact Implementation of Geometric Algorithms
, 2001
"... This thesis has two main parts. The first part deals with the problem of curve reconstruction. Given a finite sample set S from an unknown collection of curves #, the task is to compute the graph G(S,#) which has vertex set S and an edge between exactly those pairs of vertices that are adjacent on s ..."
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This thesis has two main parts. The first part deals with the problem of curve reconstruction. Given a finite sample set S from an unknown collection of curves #, the task is to compute the graph G(S,#) which has vertex set S and an edge between exactly those pairs of vertices that are adjacent on some curve in #. We present a purely combinatorial algorithm that solves the curve reconstruction problem in polynomial time. It is the first algorithm which provably handles collections of curves with corners and endpoints. In the second
Author manuscript, published in "Library Centric Software Design (LCSD) (2006)" A Generic Lazy Evaluation Scheme for Exact Geometric Computations
, 2008
"... We present a generic C++ design to perform efficient and exact geometric computations using lazy evaluations. Exact geometric computations are critical for the robustness of geometric algorithms. Their efficiency is also critical for most applications, hence the need for delaying the exact computati ..."
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We present a generic C++ design to perform efficient and exact geometric computations using lazy evaluations. Exact geometric computations are critical for the robustness of geometric algorithms. Their efficiency is also critical for most applications, hence the need for delaying the exact computations at run time until they are actually needed. Our approach is generic and extensible in the sense that it is possible to make it a library which users can extend to their own geometric objects or primitives. It involves techniques such as generic functor adaptors, dynamic polymorphism, reference counting for the management of directed acyclic graphs and exception handling for detecting cases where exact computations are needed. It also relies on multiple precision arithmetic as well as interval arithmetic. We apply our approach to the whole geometric kernel of Cgal.