Arrangements on Parametric Surfaces I: General Framework and Infrastructure

Abstract

Abstract. We introduce a framework for the construction, maintenance, and manipulation of arrangements of curves embedded on certain two-dimensional orientable parametric surfaces in three-dimensional space. The framework applies to planes, cylinders, spheres, tori, and surfaces homeomorphic to them. We reduce the effort needed to generalize existing algorithms, such as the sweep line and zone traversal algorithms, originally designed for arrangements of bounded curves in the plane, by extensive reuse of code. We have realized our approach as the Cgal package Arrangement on surface 2. 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.

Arrangements on Parametric Surfaces II: Concretizations and Applications

“…• We have to deal with identifications on both pais of opposite sides of the boundary, that is, we have to provide comparisons of curve-ends near the boundaries, to check whether a point or curve lies on an identification, and to compare points on identified sides; see [8] for more details. Observe that the Curved kernel via analysis 2 is a model that deals with four open (unbounded in parameter space) sides and that the comparisons near the boundaries still perfectly fit.…”

“…Our topology-traits class minds the case that the first non-contractible closed curve does not result in a face split, but only convert the torus-like initial face into an cylinder-like face. For more details on this issue we refer the reader to [8].…”

“…We divide the presented classes into two categories distinguished by the type of arithmetic required for their implementation, as the type of arithmetic plays a significant role in exact computation. The classes of arrangements we present in this paper are concretizations of a new generic framework [7,8]. Relying on such a framework in general is advantageous for two reasons: (i) new concretizations are relatively easily produced, and (ii) all functionality available for the framework immediately becomes available for the new concretization.…”

“…Arrangements in general are fundamental data-structures in computational geometry, and have been intensively studied for several decades. For related work on arrangements see Section 1 of the companion paper [8] and the references therein. Arrangements of linear objects in the plane in particular have many theoretical and practical applications (see, e. g., [2,17,31,38]), and many of them have analogous applications, where the embedding plane and the inducing linear curves are substituted for the sphere and geodesic arcs on it.…”

“…Outline. For the rest of the article we assume the reader to be familiar with the framework introduced in the companion paper [8]; especially with its notion and the geometry-traits and topology-traits concepts. We provide implementation details and application samples of our first non-planar concretizations: Section 2 covers geodesic arcs on the sphere and Section 3 presents arrangements on ring Dupin cyclides and elliptic quadrics.…”