On-line Zone Construction in Arrangements of Lines in the Plane

Given a finite set L of lines in the plane we wish to compute the zone of an additional curve fl in the arrangement A(L), namely the set of faces of the planar subdivision induced by the lines in L that are crossed by fl, where fl is not given in advance but rather provided on-line 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). We implemented the four algorithms. We present implementation details, comparison of performance, and a discussion of the advantages and shortcomings of each of the proposed algorithms. 1 Introduction Given a finite collection L of lines in the plane, the arrangement A(L) is the subdivision of the plane into vertices, edges and faces induced by L. Arrangements of lines in the plane, as well as arrangements of other objects and in higher dimensional spaces, ..

Code flexibility and program efficiency by genericity: Improving cgal’s arrangements

Abstract. Arrangements of planar curves are fundamental structures in computational geometry. We describe the recent developments in the arrangement package of Cgal, the Computational Geometry Algorithms Library, making it easier to use, to extend and to adapt to a variety of applications. This improved flexibility of the code does not come at the expense of efficiency as we mainly use generic-programming techniques, which make dexterous use of the compilation process. To the contrary, we expedited key operations as we demonstrate by experiments.