Arrangements on Parametric Surfaces II: Concretizations and Applications

Abstract

Abstract. We describe the algorithms and implementation details involved in the concretizations of a generic framework that enables exact construction, maintenance, and manipulation of arrangements embedded on certain two-dimensional orientable parametric surfaces in three-dimensional space. The fundamentals of the framework are described in a companion paper. Our work covers arrangements embedded on elliptic quadrics and cyclides induced by intersections with other algebraic surfaces, and a specialized case of arrangements induced by arcs of great circles embedded on the sphere. We also demonstrate how such arrangements can be used to accomplish various geometric tasks efficiently, such as computing the Minkowski sums of polytopes, the envelope of surfaces, and Voronoi diagrams embedded on parametric surfaces. We do not assume general position. Namely, we handle degenerate input, and produce exact results in all cases. Our implementation is realized using Cgal and, in particular, the package that provides the underlying framework. We have conducted experiments on various data sets, and documented the practical efficiency of our approach.Mathematics Subject Classification (2010). Primary 68U05; Secondary 14Q10.

On the Power of Manifold Samples in Exploring Configuration Spaces and the Dimensionality of Narrow Passages

“…This means in particular that M will consist of manifolds of dimension 4 d 2 . With this prerequisite in mind, there is already what to gain from using our existing and strong machinery for analyzing two-dimensional manifolds [4,5,11], while fulfilling the conditions above: We can solve motionplanning problems with four degrees of freedom, at the strength level that MMS offers, which is higher than that of standard sampling-based tools.…”

Arrangements on Parametric Surfaces I: General Framework and Infrastructure

“…The companion paper [6] discusses arrangements on spheres, quadrics, and ring Dupin cyclides. The Arrangement on surface 2 package contains two topology-traits classes for the plane: one for bounded curves, which maintains a single unbounded face, and one for the unbounded plane, which uses an implicit bounding rectangle; see Section 3.2.4 and [31] for more details.…”

“…This subdivision is the arrangement A(C) induced by C on S. We present a generic framework for the construction, maintenance, and manipulation of arrangements embedded on two-dimensional orientable parametric surfaces such 2 Berberich, Fogel, Halperin, Mehlhorn and Wein as planes, cylinders, spheres, tori, and surfaces homeomorphic to them. Such arrangements have many theoretical and practical applications [1,6,16,20]. Our work is conceptual -the definition of the framework -and practical -the implementation of the framework.…”

“…The algorithmic machinery is generic and provided by the package. We and others have already instantiated the package for different surfaces and curves; see Section 6 and the companion paper [6]. Examples are arrangement of lines in the plane, of arcs of great circles on the sphere [17,18], of intersection curves between quadric surfaces and a fixed quadric [7], and of intersection curves between arbitrary algebraic surfaces and a fixed Dupin cyclide [9].…”

“…We define a compact interface for our framework; only the operations in the interface need to be implemented for a specific application. The companion paper [6] describes concretizations for several types of surfaces and curves embedded on them, and applications. This is the first implementation of a generic algorithm that can handle arrangements on a large class of parametric surfaces.…”