Exact, efficient, and complete arrangement computation for cubic curves

Abstract

The Bentley-Ottmann sweep-line 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.

Improving the topology computation of an arrangement of cubics

“…The proposed improvement comes from using deeper the well-known geometry of the reducible cubics instead of relying on general algebraic tools. This idea has also been used in [3] to extend the algorithm in [6] for analyzing arrangements of cubic curves to study in a similar way the topology of an arrangement of quartic curves.…”

“…In this case, the considered cubic is known to be reducible and this fact is used to simplify many of the calculations proposed in [6]. The proposed improvement comes from using deeper the well-known geometry of the reducible cubics instead of relying on general algebraic tools.…”

“…After presenting in the first section some algebraic and geometric preliminaries devoted to polynomials, subresultants and the geometry of cubic curves, the second section briefly reviews the algorithm in [6] paying special attention to the part of the algorithm we are going to improve. The third section contains the detailed description of our proposal for the analysis of a single reducible cubic and the last section presents several examples together with a report of the performed experimentation showing how the proposed modification improves the overall efficiency of the algorithm introduced in [6].…”

“…Efficient algorithms for arrangements of straight segments can be found in [12], [7] and [13] and for conics in [14] and [1]. In [6] the authors introduced a complete, exact and efficient algorithm for computing the topology of an arrangement of cubic curves by adapting the Bentley-Ottmann sweep-line method to this situation through the use of limited algebraic machinery (mainly resultants, subresultants and real root determination for univariate polynomials). This algorithm proceeds by analyzing first every single cubic, then by studying every pair of cubics in the arrangement and, finally, by merging all the available information.…”

“…The main goal of this paper is a complete modification of the analysis of a single cubic in the algorithm introduced in [6] when the considered cubic has at least two singularities. In this case, the considered cubic is known to be reducible and this fact is used to simplify many of the calculations proposed in [6].…”