372 research outputs found
Extremal properties for dissections of convex 3-polytopes
A dissection of a convex d-polytope is a partition of the polytope into
d-simplices whose vertices are among the vertices of the polytope.
Triangulations are dissections that have the additional property that the set
of all its simplices forms a simplicial complex. The size of a dissection is
the number of d-simplices it contains. This paper compares triangulations of
maximal size with dissections of maximal size. We also exhibit lower and upper
bounds for the size of dissections of a 3-polytope and analyze extremal size
triangulations for specific non-simplicial polytopes: prisms, antiprisms,
Archimedean solids, and combinatorial d-cubes.Comment: 19 page
Decompositions of a polygon into centrally symmetric pieces
In this paper we deal with edge-to-edge, irreducible decompositions of a
centrally symmetric convex -gon into centrally symmetric convex pieces.
We prove an upper bound on the number of these decompositions for any value of
, and characterize them for octagons.Comment: 17 pages, 17 figure
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An Ontology for Grounding Vague Geographic Terms
Many geographic terms, such as âriverâ and âlakeâ, are vague, with no clear boundaries of application. In particular, the spatial extent of such features is often vaguely carved out of a continuously varying observable domain. We present a means of defining vague terms using standpoint semantics, a refinement of the
philosophical idea of supervaluation semantics. Such definitions can be grounded in actual data by geometric analysis and segmentation of the data set. The issues
raised by this process with regard to the nature of boundaries and domains of logical quantification are discussed. We describe a prototype implementation of a system capable of segmenting attributed polygon data into geographically significant regions and evaluating queries involving vague geographic feature terms
On Dissecting Polygons into Rectangles
What is the smallest number of pieces that you can cut an n-sided regular
polygon into so that the pieces can be rearranged to form a rectangle? Call it
r(n). The rectangle may have any proportions you wish, as long as it is a
rectangle. The rules are the same as for the classical problem where the
rearranged pieces must form a square. Let s(n) denote the minimum number of
pieces for that problem. For both problems the pieces may be turned over and
the cuts must be simple curves. The conjectured values of s(n), 3 <= n <= 12,
are 4, 1, 6, 5, 7, 5, 9, 7, 10, 6. However, only s(4)=1 is known for certain.
The problem of finding r(n) has received less attention. In this paper we give
constructions showing that r(n) for 3 <= n <= 12 is at most 2, 1, 4, 3, 5, 4,
7, 4, 9, 5, improving on the bounds for s(n) in every case except n=4. For the
10-gon our construction uses three fewer pieces than the bound for s(10). Only
r(3) and r(4) are known for certain. We also briefly discuss q(n), the minimum
number of pieces needed to dissect a regular n-gon into a monotile.Comment: 26 pages, one table, 41 figures, 14 reference
Hinged Dissections Exist
We prove that any finite collection of polygons of equal area has a common
hinged dissection. That is, for any such collection of polygons there exists a
chain of polygons hinged at vertices that can be folded in the plane
continuously without self-intersection to form any polygon in the collection.
This result settles the open problem about the existence of hinged dissections
between pairs of polygons that goes back implicitly to 1864 and has been
studied extensively in the past ten years. Our result generalizes and indeed
builds upon the result from 1814 that polygons have common dissections (without
hinges). We also extend our common dissection result to edge-hinged dissections
of solid 3D polyhedra that have a common (unhinged) dissection, as determined
by Dehn's 1900 solution to Hilbert's Third Problem. Our proofs are
constructive, giving explicit algorithms in all cases. For a constant number of
planar polygons, both the number of pieces and running time required by our
construction are pseudopolynomial. This bound is the best possible, even for
unhinged dissections. Hinged dissections have possible applications to
reconfigurable robotics, programmable matter, and nanomanufacturing.Comment: 22 pages, 14 figure
Meshes Preserving Minimum Feature Size
The minimum feature size of a planar straight-line graph is the minimum distance between a vertex and a nonincident edge. When such a graph is partitioned into a mesh, the degradation is the ratio of original to final minimum feature size. For an n-vertex input, we give a triangulation (meshing) algorithm that limits degradation to only a constant factor, as long as Steiner points are allowed on the sides of triangles. If such Steiner points are not allowed, our algorithm realizes \ensuremathO(lgn) degradation. This addresses a 14-year-old open problem by Bern, Dobkin, and Eppstein
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