George M. Tzoumas

The predicates for the Voronoi diagram of ellipses

This paper examines the computation of the Voronoi diagram of a set of ellipses in the Euclidean plane. We propose the first complete algorithms, under the exact computation paradigm, for the predicates of an incremental algorithm: κ1 decides which one of 2 given ellipses is closest to a given exterior point; κ2 decides the position of a query ellipse relative to an external bitangent line of 2 given ellipses; κ3 decides the position of a query ellipse relative to a Voronoi circle of 3 given ellipses; κ4 determines the type of conflict between a Voronoi edge, defined by 4 given ellipses, and a query ellipse. The paper is restricted to non-intersecting ellipses, but the extension to arbitrary ones is possible.The ellipses are input in parametric representation or constructively in terms of their axes, center and rotation. For κ1 and κ2 we derive optimal algebraic conditions, solve them exactly and provide efficient implementations in C++. For κ3 we compute a tight bound on the number of complex tritangent circles and use the parametric form of the ellipses in order to design an exact subdivision-based algorithm, which is implemented on Maple. This approach essentially answers κ4 as well. We conclude with current work on optimizing κ3 and implementing it in C++.

THE PREDICATES FOR THE EXACT VORONOI DIAGRAM OF ELLIPSES UNDER THE EUCLIDIEAN METRIC

This article examines the computation of the Delaunay graph and its dual Voronoi diagram of a set of ellipses in the Euclidean plane. We propose the first complete methods, under the exact computation paradigm, for the predicates of an incremental algorithm: κ1 decides which one of two given ellipses is closest to a given exterior point; κ2 decides the position of a query ellipse relative to an external bitangent line of two given ellipses; κ3 decides the position of a query ellipse relative to a Voronoi circle of three given ellipses; κ4 determines the type of conflict between a Voronoi edge, defined by four given ellipses, and a query ellipse. The article is restricted to non-intersecting ellipses, but the extension to arbitrary ones is possible. The ellipses are input in parametric representation, i.e., constructively in terms of their axes, center and rotation. For κ1 and κ2 we derive algebraic conditions optimal in terms of the degree of the algebraic numbers involved, and provide efficient implementations in C++. For κ3 we compute a tight bound on the number of complex tritangent circles and design an exact symbolic-numeric algorithm, which is implemented in MAPLE. This essentially answers κ4 as well. We conclude with further work on lifting the condition of non-intersecting ellipses.

Exact Delaunay graph of smooth convex pseudo-circles: general predicates, and implementation for ellipses

We examine the problem of computing exactly the Delaunay graph (and the dual Voronoi diagram) of a set of, possibly intersecting, smooth convex pseudo-circles in the Euclidean plane, given in parametric form. Pseudo-circles are (convex) sites, every pair of which has at most two intersecting points. The Delaunay graph is constructed incrementally. Our first contribution is to propose robust end efficient algorithms for all required predicates, thus generalizing our earlier algorithms for ellipses, and we analyze their algebraic complexity, under the exact computation paradigm. Second, we focus on InCircle, which is the hardest predicate, and express it by a simple sparse 5 X 5 polynomial system, which allows for an efficient implementation by means of successive Sylvester resultants and a new factorization lemma. The third contribution is our cgal-based c++ software for the case of ellipses, which is the first exact implementation for the problem. Our code spends about 98 sec to construct the Delaunay graph of 128 non-intersecting ellipses, when few degeneracies occur. It is faster than the cgal segment Delaunay graph, when ellipses are approximated by k-gons for k > 15.

Exact and efficient evaluation of the InCircle predicate for parametric ellipses and smooth convex objects

We study the Voronoi diagram, under the Euclidean metric, of a set of ellipses, given in parametric representation. The article concentrates on the InCircle predicate, which is the hardest to compute, and describes an exact and complete solution. It consists of a customized subdivision-based method that achieves quadratic convergence, leading to a real-time implementation for non-degenerate inputs. Degenerate cases are handled using exact algebraic computation. We conclude with experiments showing that most instances run in less than 0.1 s, on a 2.6 GHz Pentium-4, whereas degenerate cases may take up to 13 s. Our approach readily generalizes to smooth convex objects.

A real-time and exact implementation of the predicates for the Voronoi diagram of parametric ellipses

We study the Voronoi diagram, under the Euclidean metric, of a set of ellipses, given in parametric representation. We use an efficient incremental algorithm and focus on the required predicates. The paper concentrates on InCircle, which is the hardest predicate: it decides the position of a query ellipse relative to the Voronoi circle of three given ellipses. We describe an exact, real-time, and complete implementation for InCircle, combining a certified numeric algorithm with algebraic computation. The numeric part leads to a real-time implementation for non-degenerate inputs. It relies on a geometric preprocessing that guarantees a unique solution in a box of parametric space, where a customized subdivision-based method approximates the Voronoi circle tracing the bisectors. Our subdivision method achieves quadratic convergence by exploiting the geometric characteristics of the problem. To achieve robustness, we develop interval-arithmetic techniques, based on the C++ package Alias. We switch to an algebraic approach for handling the degeneracies fast. Based on a different algebraic system to model InCircle, we apply real solving and resultant theory. The latter relies on certain symbolic routines which are efficiently implemented in Maple. Our approach readily generalizes to arbitrary conics. The paper concludes with experiments showing that most instances run in less than 0.1 sec, on a 2.6GHz Pentium-4, whereas degenerate cases may take up to 13 sec.

Exact Voronoi diagram of smooth convex pseudo-circles: General predicates, and implementation for ellipses

We examine the problem of computing exactly the Voronoi diagram (via the dual Delaunay graph) of a set of, possibly intersecting, smooth convex pseudo-circles in the Euclidean plane, given in parametric form. Pseudo-circles are (convex) sites, every pair of which has at most two intersecting points. The Voronoi diagram is constructed incrementally. Our first contribution is to propose robust and efficient algorithms, under the exact computation paradigm, for all required predicates, thus generalizing earlier algorithms for non-intersecting ellipses. Second, we focus on InCircle, which is the hardest predicate, and express it by a simple sparse 5x5 polynomial system, which allows for an efficient implementation by means of successive Sylvester resultants and a new factorization lemma. The third contribution is our cgal-based c++ software for the case of possibly intersecting ellipses, which is the first exact implementation for the problem. Our code spends about a minute to construct the Voronoi diagram of 200 ellipses, when few degeneracies occur. It is faster than the cgal segment Voronoi diagram, when ellipses are approximated by k-gons for k>15, and a state-of-the-art implementation of the Voronoi diagram of points, when each ellipse is approximated by more than 1250 points.

Extending CSG with projections

We extend traditional Constructive Solid Geometry (CSG) trees to support the projection operator. Existing algorithms in the literature prove various topological properties of CSG sets. Our extension readily allows these algorithms to work on a greater variety of sets, in particular parametric sets, which are extensively used in CAD/CAM systems. Constructive Solid Geometry allows for algebraic representation which makes it easy for certification tools to apply. A geometric primitive may be defined in terms of a characteristic function, which can be seen as the zero-set of a corresponding system along with inequality constraints. To handle projections, we exploit the Disjunctive Normal Form, since projection distributes over union. To handle intersections, we transform them into disjoint unions. Each point in the projected space is mapped to a contributing primitive in the original space. This way we are able to perform gradient computations on the boundary of the projected set through equivalent gradient computations in the original space. By traversing the final expression tree, we are able to automatically generate a set of equations and inequalities that express either the geometric solid or the conditions to be tested for computing various topological properties, such as homotopy equivalence. We conclude by presenting our prototype implementation and several examples. Extension of classical CSG with the projection operator.Support for gradient computations.Topological property computation.Application of formal methods like interval analysis and proof assistants to CSG models.

Data structures and algorithms for topological analysis

One of the steps of geometric modeling is to know the topology and/or the geometry of the objects considered. This paper presents different data structures and algorithms used in this study. We are particularly interested by algebraic structures, eg homotopy and homology groups, the Betti numbers, the Euler characteristic, or the Morse-Smale complex. We have to be able to compute these data structures, and for (homotopy and homology) groups, we also want to compute their generators. We are also interested in algorithms CIA and HIA presented in the thesis of Nicolas DELANOUE, which respectively compute the connected components and the homotopy type of a set defined by a CSG (constructive solid geometry) tree. We would like to generalize these algorithms to sets defined by projection.

Apollonius circle conflict

What does a query ellipse do (outside, intersects or tangent) with respect to the Apollonius circle of the three black ellipses?