State of the Art: Updating Delaunay Triangulations for Moving Points

This paper considers the problem of updating efficiently a two-dimensional Delaunay triangulation when vertices are moving. We investigate the three current state-of-the-art approaches to solve this problem: --1-- the use of kinetic data structures, --2-- the possibility of moving points from their initial to final position by deletion and insertion and --3-- the use of "almost" Delaunay structure that postpone the necessary modifications. Finally, we conclude with a global overview of the above-mentioned approaches while focusing on future works

Part of the Computer Sciences Commons Comments Victor Milenkovic & Elisha Sacks

We present two approximate Minkowski sum algorithms for planar regions bounded by line and circle segments. Both algorithms form a convolution curve, construct its arrangement, and use winding numbers to identify sum cells. The first uses the kinetic convolution and the second uses our monotonic convolution. The asymptotic running times of the exact algorithms are increased by km log m with m the number of segments in the convolution and with k the number of segment triples that are in cyclic vertical order due to approximate segment intersection. The approximate Minkowski sum is close to the exact sum of perturbation regions that are close to the input regions. We validate both algorithms on part packing tasks with industrial part shapes. The accuracy is near the floating point accuracy even after multiple iterated sums. The programs are 2% slower than direct floating point implementations of the exact algorithms. The monotonic algorithm is 42% faster than the kinetic algorithm

SYNAPS: A library for dedicated applications in symbolic numeric computing,

International audienceWe present an overview of the open source library synaps. We describe some of the representative algorithms of the library and illustrate them on some explicit computations, such as solving polynomials and computing geometric information on implicit curves and surfaces. Moreover, we describe the design and the techniques we have developed in order to handle a hierarchy of generic and specialized data-structures and routines, based on a view mechanism. This allows us to construct dedicated plugins, which can be loaded easily in an external tool. Finally, we show how this design allows us to embed the algebraic operations, as a dedicated plugin, into the external geometric modeler axel

AN APPROXIMATE ARRANGEMENT ALGORITHM FOR SEMI-ALGEBRAIC CURVES

We present an arrangement algorithm for plane curves. The inputs are (1) continuous, compact, x-monotone curves and (2) a module that computes approximate crossing points of these curves. There are no general position requirements. We assume that the crossing module output is ǫ accurate, but allow it to be inconsistent, meaning that three curves are in cyclic y order over an x interval. The curves are swept with a vertical line using the crossing module to compute and process sweep events. When the sweep detects an inconsistency, the algorithm breaks the cycle to obtain a linear order. We prove correctness in a realistic computational model of the crossing module. The number of vertices in the output is V = 2n+N+min(3kn, n 2 /2) and the running time is O(V log n) for n curves with N crossings and k inconsistencies. The output arrangement is realizable by curves that are O(ǫ + knǫ) close to the input curves, except in knǫ neighborhoods of the curve tails. The accuracy can be guaranteed everywhere by adding tiny horizontal extensions to the segment tails, but without the running time bound. An implementation is described for semi-algebraic curves based on a numerical equation solver. Experiments show that the extensions only slightly increase the running time and have little effect on the error. On challenging data sets, the number of inconsistencies is at most 3N, the output accuracy is close to ǫ, and the running time is close to that of the standard, non-robust floating point sweep

Geometric algorithms for algebraic curves and surfaces

This work presents novel geometric algorithms dealing with algebraic curves and surfaces of arbitrary degree. These algorithms are exact and complete — they return the mathematically true result for all input instances. Efficiency is achieved by cutting back expensive symbolic computation and favoring combinatorial and adaptive numerical methods instead, without spoiling exactness in the overall result. We present an algorithm for computing planar arrangements induced by real algebraic curves. We show its efficiency both in theory by a complexity analysis, as well as in practice by experimental comparison with related methods. For the latter, our solution has been implemented in the context of the Cgal library. The results show that it constitutes the best current exact implementation available for arrangements as well as for the related problem of computing the topology of one algebraic curve. The algorithm is also applied to related problems, such as arrangements of rotated curves, and arrangments embedded on a parameterized surface. In R3, we propose a new method to compute an isotopic triangulation of an algebraic surface. This triangulation is based on a stratification of the surface, which reveals topological and geometric information. Our implementation is the first for this problem that makes consequent use of numerical methods, and still yields the exact topology of the surface.Diese Arbeit stellt neue Algorithmen für algebraische Kurven und Flächen von beliebigem Grad vor. Diese Algorithmen liefern für alle Eingaben das mathematisch korrekte Ergebnis. Wir erreichen Effizienz, indem wir aufwendige symbolische Berechnungen weitesgehend vermeiden, und stattdessen kombinatorische und adaptive numerische Methoden einsetzen, ohne die Exaktheit des Resultats zu zerstören. Der Hauptbeitrag ist ein Algorithmus zur Berechnung von planaren Arrangements, die durch reelle algebraische Kurven induziert sind. Wir weisen die Effizienz des Verfahrens sowohl theoretisch durch eine Komplexitätsanalyse, als auch praktisch durch experimentelle Vergleiche nach. Dazu haben wir unser Verfahren im Rahmen der Softwarebibliothek Cgal implementiert. Die Resultate belegen, dass wir die zur Zeit beste verfügbare exakte Software bereitstellen. Der Algorithmus wird zur Arrangementberechnung rotierter Kurven, oder für Arrangements auf parametrisierten Oberflächen eingesetzt. Im R3 geben wir ein neues Verfahren zur Berechnung einer isotopen Triangulierung einer algebraischen Oberfläche an. Diese Triangulierung basiert auf einer Stratifizierung der Oberfläche, die topologische und geometrische Informationen berechnet. Unsere Implementierung ist die erste für dieses Problem, welche numerische Methoden konsequent einsetzt, und dennoch die exakte Topologie der Oberfläche liefert