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Minkowski Addition of Triangles

Abstract : Geometric modelling of a mixture management manufacturing problem (simultaneous feasibility of two mixtures) yields new polytopes, resulting from the sum of particular triangles, which are called "2- mixtures convex sets" in this context. More generally, the sum of triangles can be considered as a generalization of zonotopes (sum of segments). From this point of view, this study shows that the zone property associated to a zonotope segment is generalized to three "half-zones" associated to each triangle. The combinatorial complexity of these polytopes (the number of faces of the polytopes) with respect to the number of summands, has the same order as that of zonotopes. We also study how to construct such polytopes, and we propose algorithms that are optimal in time. Concerning the particular problem of mixture management, the first non trivial case is that of three components mixtures, which lies in six-dimensional space. Therefore, if a point belongs to the "2-mixtures convex set", then the mixtures are simultaneously feasible. We describe the facets of this polytope in six dimensional space, in order to obtain feasibility conditions for two mixtures. Finally, we study the decomposition of polytopes and we expose main kwon results in this field.
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Submitted on : Monday, February 23, 2004 - 5:59:12 PM
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  • HAL Id : tel-00005017, version 1

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Mireille Rousset. Minkowski Addition of Triangles. Modélisation et simulation. Université Joseph-Fourier - Grenoble I, 1996. Français. ⟨tel-00005017⟩

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