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- Beldiceanu, Nicolas, et al.
(författare)
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Non-overlapping Constraints between Convex Polytopes
- 2001. - 1
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- This paper deals with non-overlapping constraints between convex polytopes. Non-overlapping detection between fixed objects is a fundamental geometric primitive that arises in many applications. However from a constraint perspective it is natural to extend the previous problem to a non-overlapping constraint between two objects for which both positions are not yet fixed. A first contribution is to present theorems for convex polytopes which allow coming up with general necessary conditions for non-overlapping. These theorems can be seen as a generalization of the notion of compulsory part which was introduced in 1984 by Lahrichi and Gondran [6] for managing non-overlapping constraint between rectangles. Finally, a second contribution is to derive from the previous theorems efficient filtering algorithms for two special cases: the non-overlapping constraint between two convex polygons as well as the non-overlapping constraint between d-dimensional boxes.
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| 2. |
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| 3. |
- Guo, Qi, 1957-
(författare)
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Minkowski Measure of Asymmetry and Minkowski Distance for Convex Bodies
- 2004
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis consists of four papers about the Minkowski measure of asymmetry and the Minkowski (or Banach-Mazur) distance for convex bodies.We relate these two quantities by giving estimates for the Minkowski distance in terms of the Minkowski measure. We also investigate some properties of the Minkowski measure, in particular a stability estimate is given. More specifically, let C and D be n-dimensional convex bodies. Denote by As(C) and As(D) the Minkowski measures of asymmetry of C and D resp. and by d(C,D) the Minkowski distance between C and D.In Paper I, by using a linearisation method for affine spaces and affine maps and using a generalisation of a lemma of D.R. Lewis, we proved that d(C,D) < n(As(C) + As(D))/2 for all convex bodies C,D.In Paper II, by first proving some general existence theorems for a class of volume-increasing affine maps, we obtain the estimate that under the same conditions as in paper I, d(C,D) < (n-1) min(As(C),As(D)) + n.In Paper III we consider the Minkowski measure itself. We determine the Minkowski measures for convex hulls of sets of the form conv(C,p) where C is a convex set with known measure of asymmetry and p is a point outside C.In Paper IV, we focus on estimating the deviation of a convex body C from the simplex S if the Minkowski measure of C is close to the maximum value n (known to be attained only for the simplex). We prove that if As(C) > n - ε for 0 < ε < 1/δ where δ = 8(n+1), then d(C,S) < 1 + 8(n+1) ε .
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| 4. |
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| 5. |
- Kaijser, Sten, et al.
(författare)
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Asymmetry of Some Convex Bodies
- 2002
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Ingår i: DiscreteComput. Geom.. ; 27, s. 239-247
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Tidskriftsartikel (refereegranskat)
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