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**1 - 4**of**4**### The predicates of the Apollonius diagram: algorithmic analysis and implementation

"... We study the predicates involved in an efficient dynamic algorithm for computing the Apollonius diagram in the plane, also known as the additively weighted Voronoi diagram. We present a complete algorithmic analysis of these predicates, some of which are reduced to simpler and more easily computed p ..."

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We study the predicates involved in an efficient dynamic algorithm for computing the Apollonius diagram in the plane, also known as the additively weighted Voronoi diagram. We present a complete algorithmic analysis of these predicates, some of which are reduced to simpler and more easily computed primitives. This gives rise to an exact and efficient implementation of the algorithm, that handles all special cases. Among our tools we distinguish an inversion transformation and an infinitesimal perturbation for handling degeneracies. The implementation of the predicates requires certain algebraic operations. In studying the latter, we aim at minimizing the algebraic degree of the predicates and the number of arithmetic operations; this twofold optimization corresponds to reducing bit complexity. The proposed algorithms are based on static Sturm se-quences. Multivariate resultants provide a deeper understanding of the predicates and are compared against our methods. We expect that our algebraic techniques are sufficiently powerful and general to be applied to a number of analogous geometric problems on curved objects. Their efficiency, and that of the overall implementation, are illustrated by a series of numerical experiments. Our approach can be imme-diately extended to the incremental construction of abstract Voronoi diagrams for various classes of objects. Key words: computational geometry, algebraic computing, Apollonius diagram, geometric predicates, Sturm sequence, resultant

### CONSTRUCTING THE VISIBILITY COMPLEX OF POLYTOPES IN 3-SPACE

"... Abstract. We prove the connexity of the boundary of each 4-dimensional cell of the 3D visibility complex of disjoint convex polyhedra. Using this property, we outline a simple algorithm for the construction of the visibility complex of k disjoint, convex polyhedra in time O(n 2 k 2 log n) where n is ..."

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Abstract. We prove the connexity of the boundary of each 4-dimensional cell of the 3D visibility complex of disjoint convex polyhedra. Using this property, we outline a simple algorithm for the construction of the visibility complex of k disjoint, convex polyhedra in time O(n 2 k 2 log n) where n is the total complexity of the polyhedra. 1.

### unknown title

"... We show how to compute the bitangents of two ellipses, and how to evaluate the predicates required to compute objects like convex hulls or visibility complexes for ellipses. Also, we show how to compute the tangency points of the bitangents. ..."

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We show how to compute the bitangents of two ellipses, and how to evaluate the predicates required to compute objects like convex hulls or visibility complexes for ellipses. Also, we show how to compute the tangency points of the bitangents.