Classroom examples of robustness problems in geometric computations

Kettner, Lutz; Mehlhorn, Kurt; Pion, Sylvain; Schirra, Stefan; Yap, Chee

doi:10.1016/j.comgeo.2007.06.003

Cited by 90 publications

(40 citation statements)

References 20 publications

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“…All numeric computations are described using "abstract perfect" real numbers. In practice, specialists in algorithmic geometry know that numeric computation with floating point numbers can incur failures of the algorithm by failing to detect illegal edges, or by giving inconsistent results for several related computations [33,24]. For instance, rounding errors could make that both an edge and its flipped counterpart could appear to be illegal, thus leading to looping computation that is not predicted by our ideal formal model.…”

Section: Resultsmentioning

confidence: 99%

Formal Study of Plane Delaunay Triangulation

Dufourd

Bertot

2010

Interactive Theorem Proving

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This article presents the formal proof of correctness for a plane Delaunay triangulation algorithm. It consists in repeating a sequence of edge flippings from an initial triangulation until the Delaunay property is achieved. To describe triangulations, we rely on a combinatorial hypermap specification framework we have been developing for years. We embed hypermaps in the plane by attaching coordinates to elements in a consistent way. We then describe what are legal and illegal Delaunay edges and a flipping operation which we show preserves hypermap, triangulation, and embedding invariants. To prove the termination of the algorithm, we use a generic approach expressing that any non-cyclic relation is well-founded when working on a finite set.

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Section: Resultsmentioning

confidence: 99%

Formal Study of Plane Delaunay Triangulation

Dufourd

Bertot

2010

Interactive Theorem Proving

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“…Geometric algorithms often rely on predicates, i.e., estimation of finite-valued geometric quantities such as the orientation of a quadruple of points in 3D, or the number of intersections between a straight line and a surface. These predicates need to be evaluated exactly: if this is not the case, some geometric algorithm may even not terminate [12]. We adopt the dynamic filtering technique for the computation of our predicates in CGAL.…”

Section: Computing the Intersection Of A Power Diagram With A Spherementioning

confidence: 99%

Far-field reflector problem and intersection of paraboloids

2015

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International audienceIn this article, we study the intersection (or union) of the convex hull of N confocal paraboloids (or ellipsoids) of revolution. This study is motivated by a Minkowski-type problem arising in geometric optics. We show that in each of the four cases, the combinatorics is given by the intersection of a power diagram with the unit sphere. We prove the complexity is O(N) for the intersection of paraboloids and Omega(N^2) for the intersection and the union of ellipsoids. We provide an algorithm to compute these intersections using the exact geometric computation paradigm. This algorithm is optimal in the case of the intersection of ellipsoids and is used to solve numerically the far-field reflector problem

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“…Implementing geometric algorithms is notoriously dicult, especially because of numerical issues. Although geometric algorithms are basically of combinatorial and discrete nature, the branching decisions are based upon continuous predicate evaluations, subject to rounding errors when oating point arithmetic is used, which often produces inconsistencies [31].…”

Section: Predicates Versus Constructions In the Exact Geometric Compumentioning

confidence: 99%