TY - JOUR

T1 - Arrangements on Parametric Surfaces I

T2 - GeneralFramework and Infrastructure

AU - Berberich, Eric

AU - Fogel, Efi

AU - Halperin, Dan

AU - Mehlhorn, Kurt

AU - Wein, Ron

N1 - Funding Information:
This work has been supported in part by the Israel Science Foundation (grant no. 236/06), by the German–Israeli Foundation (grant no. 969/07), and by the Hermann Minkowski–Minerva Center for Geometry at Tel Aviv University. A preliminary version of this paper [8] has appeared in the Proceedings of the 15th Annual European Symposium on Algorithms (ESA 2007).

PY - 2010/11

Y1 - 2010/11

N2 - 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 and modular interface for our framework; for a given application a required small subset of the interface can be identified. Then, only this subset must be implemented. A companion paper 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.

AB - 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 and modular interface for our framework; for a given application a required small subset of the interface can be identified. Then, only this subset must be implemented. A companion paper 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.

KW - Arrangement of curves

KW - Cgal

KW - Computational geometry

KW - Parametric surface

KW - Robust geometric computing

UR - http://www.scopus.com/inward/record.url?scp=77957778035&partnerID=8YFLogxK

U2 - 10.1007/s11786-010-0042-5

DO - 10.1007/s11786-010-0042-5

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AN - SCOPUS:77957778035

SN - 1661-8270

VL - 4

SP - 45

EP - 66

JO - Mathematics in Computer Science

JF - Mathematics in Computer Science

IS - 1

ER -