520 research outputs found
Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere
We present two exact implementations of efficient output-sensitive algorithms
that compute Minkowski sums of two convex polyhedra in 3D. We do not assume
general position. Namely, we handle degenerate input, and produce exact
results. We provide a tight bound on the exact maximum complexity of Minkowski
sums of polytopes in 3D in terms of the number of facets of the summand
polytopes. The algorithms employ variants of a data structure that represents
arrangements embedded on two-dimensional parametric surfaces in 3D, and they
make use of many operations applied to arrangements in these representations.
We have developed software components that support the arrangement
data-structure variants and the operations applied to them. These software
components are generic, as they can be instantiated with any number type.
However, our algorithms require only (exact) rational arithmetic. These
software components together with exact rational-arithmetic enable a robust,
efficient, and elegant implementation of the Minkowski-sum constructions and
the related applications. These software components are provided through a
package of the Computational Geometry Algorithm Library (CGAL) called
Arrangement_on_surface_2. We also present exact implementations of other
applications that exploit arrangements of arcs of great circles embedded on the
sphere. We use them as basic blocks in an exact implementation of an efficient
algorithm that partitions an assembly of polyhedra in 3D with two hands using
infinite translations. This application distinctly shows the importance of
exact computation, as imprecise computation might result with dismissal of
valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages
long. The advisor was Prof. Dan Halperi
The maximum number of faces of the Minkowski sum of two convex polytopes
We derive tight expressions for the maximum number of -faces,
, of the Minkowski sum, , of two
-dimensional convex polytopes and , as a function of the number
of vertices of the polytopes.
For even dimensions , the maximum values are attained when and
are cyclic -polytopes with disjoint vertex sets. For odd dimensions
, the maximum values are attained when and are
-neighborly -polytopes, whose vertex sets are
chosen appropriately from two distinct -dimensional moment-like curves.Comment: 37 pages, 8 figures, conference version to appear at SODA 2012; v2:
fixed typos, made stylistic changes, added figure
The maximum number of faces of the Minkowski sum of three convex polytopes
We derive tight expressions for the maximum
number of -faces, , of the
Minkowski sum, , of three -dimensional convex polytopes , and in ,
as a function of the number of vertices of the polytopes, for any .
Expressing the Minkowski sum as a section of the Cayley polytope of its summands, counting the -faces of reduces to counting the -faces of which meet the vertex sets of the three polytopes.
In two dimensions our expressions reduce to known results,
while in three dimensions, the tightness of our bounds follows by exploiting known tight bounds for the number of faces of -polytopes in , where .
For , the maximum values are attained when
, and are -polytopes, whose vertex sets are chosen appropriately from three distinct -dimensional moment-like curves
Topological obstructions for vertex numbers of Minkowski sums
We show that for polytopes P_1, P_2, ..., P_r \subset \R^d, each having n_i
\ge d+1 vertices, the Minkowski sum P_1 + P_2 + ... + P_r cannot achieve the
maximum of \prod_i n_i vertices if r \ge d. This complements a recent result of
Fukuda & Weibel (2006), who show that this is possible for up to d-1 summands.
The result is obtained by combining methods from discrete geometry (Gale
transforms) and topological combinatorics (van Kampen--type obstructions) as
developed in R\"{o}rig, Sanyal, and Ziegler (2007).Comment: 13 pages, 2 figures; Improved exposition and less typos.
Construction/example and remarks adde
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