A Computational Basis for Conic Arcs and Boolean Operations on Conic Polygons

Abstract

Abstract. We give an exact geometry kernel for conic arcs, algorithms for exact computation with low-degree 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 non-negative) 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).

Exacus: Efficient and Exact Algorithms for Curves and Surfaces

“…The theory behind EXACUS is described in the following series of papers: Berberich et al [3] computed arrangements of conic arcs based on the LEDA [25] implementation of the Bentley-Ottmann sweep-line algorithm [2]. Eigenwillig et al [10] extended the sweep-line approach to cubic curves.…”

“…We based our implementation on the sweep-line algorithm for line-segments from LEDA [25], which handles all types of degeneracies in a simple unified event type. We extended the sweep-line approach with a new algorithm to reorder curve segments continuing through a common intersection point in linear time [3] and with a geometric traits class design for curve segments, which we describe in more detail here.…”

“…In particular for completeness we manufactured test data sets for all kinds of degeneracies [3,10,4]. A preliminary comparison [14] between Wein [30], Emiris et al [11], and us showed that our implementation was the only robust and stable and almost always the fastest implementation at that time.…”

“…Non-linear objects bring many new challenges with them: (1) Even single objects, e.g., a single algebraic curve, are complex, (2) there are many kinds of degeneracies, (3) and point coordinates are algebraic numbers. It is not clear yet, how to best cope with these challenges.…”

Complete, exact, and efficient computations with cubic curves

“…Exact, efficient, and complete algorithms for planar arrangements have been published by Wein [30] and Berberich et al [4] for conic segments, and by Wolpert [31] (see also [19]) for special quartic curves as part of a surface intersection algorithm. A generalization of Jacobi curves (used below for locating tangential intersections) is described by Wolpert [32].…”