843 research outputs found
Exact Minkowski sums of polyhedra and exact and efficient decomposition of polyhedra in convex pieces
We present the first exact and robust implementation of the 3D Minkowski sum of two non-convex polyhedra. Our implementation decomposes the two polyhedra into convex pieces, performs pairwise Minkowski sums on the convex pieces, and constructs their union. We achieve exactness and the handling of all degeneracies by building upon 3D Nef polyhedra as provided by Cgal. The implementation also supports open and closed polyhedra. This allows the handling of degenerate scenarios like the tight passage problem in robot motion planning. The bottleneck of our approach is the union step. We address efficiency by optimizing this step by two means: we implement an efficient decomposition that yields a small amount of convex pieces, and develop, test and optimize multiple strategies for uniting the partial sums by consecutive binary union operations. The decomposition that we implemented as part of the Minkowski sum is interesting in its own right. It is the first robust implementation of a decomposition of polyhedra into convex pieces that yields at most O(r 2) pieces, where r is the number of edges whose adjacent facets comprise an angle of more than 180 degrees with respect to the interior of the polyhedron
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
A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes
The free sum is a basic geometric operation among convex polytopes. This note
focuses on the relationship between the normalized volume of the free sum and
that of the summands. In particular, we show that the normalized volume of the
free sum of full dimensional polytopes is precisely the product of the
normalized volumes of the summands.Comment: Published in the proceedings of 2017 Southern Regional Algebra
Conferenc
Global residues for sparse polynomial systems
We consider families of sparse Laurent polynomials f_1,...,f_n with a finite
set of common zeroes Z_f in the complex algebraic n-torus. The global residue
assigns to every Laurent polynomial g the sum of its Grothendieck residues over
the set Z_f. We present a new symbolic algorithm for computing the global
residue as a rational function of the coefficients of the f_i when the Newton
polytopes of the f_i are full-dimensional. Our results have consequences in
sparse polynomial interpolation and lattice point enumeration in Minkowski sums
of polytopes.Comment: Typos corrected, reference added, 13 pages, 5 figures. To appear in
JPA
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