881 research outputs found
From affine to barycentric coordinates in polytopes
Each point of a simplex is expressed as a unique convex combination of the
vertices. The coefficients in the combination are the barycentric coordinates
of the point. For each point in a general convex polytope, there may be
multiple representations, so its barycentric coordinates are not necessarily
unique. There are various schemes to fix particular barycentric coordinates:
Gibbs, Wachspress, cartographic, etc. In this paper, a method for producing
sparse barycentric coordinates in polytopes will be discussed. It uses a purely
algebraic treatment of affine spaces and convex sets, with barycentric
algebras. The method is based on a certain decomposition of each
finite-dimensional convex polytope into a union of simplices of the same
dimension
Embedding Stacked Polytopes on a Polynomial-Size Grid
A stacking operation adds a -simplex on top of a facet of a simplicial
-polytope while maintaining the convexity of the polytope. A stacked
-polytope is a polytope that is obtained from a -simplex and a series of
stacking operations. We show that for a fixed every stacked -polytope
with vertices can be realized with nonnegative integer coordinates. The
coordinates are bounded by , except for one axis, where the
coordinates are bounded by . The described realization can be
computed with an easy algorithm.
The realization of the polytopes is obtained with a lifting technique which
produces an embedding on a large grid. We establish a rounding scheme that
places the vertices on a sparser grid, while maintaining the convexity of the
embedding.Comment: 22 pages, 10 Figure
Small grid embeddings of 3-polytopes
We introduce an algorithm that embeds a given 3-connected planar graph as a
convex 3-polytope with integer coordinates. The size of the coordinates is
bounded by . If the graph contains a triangle we can
bound the integer coordinates by . If the graph contains a
quadrilateral we can bound the integer coordinates by . The
crucial part of the algorithm is to find a convex plane embedding whose edges
can be weighted such that the sum of the weighted edges, seen as vectors,
cancel at every point. It is well known that this can be guaranteed for the
interior vertices by applying a technique of Tutte. We show how to extend
Tutte's ideas to construct a plane embedding where the weighted vector sums
cancel also on the vertices of the boundary face
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