575 research outputs found
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 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
Lifting symmetric pictures to polyhedral scenes
Scene analysis is concerned with the reconstruction of d-dimensional objects, such as polyhedral surfaces, from (d − 1)-dimensional pictures (i.e., projections of the objects onto a hyperplane). In this paper we study the impact of symmetry on the lifting properties of pictures. We first use methods from group representation theory to show that the lifting matrix of a symmetric picture can be transformed into a block-diagonalized form. Using this result we then derive new symmetry-extended counting conditions for a picture with a non-trivial symmetry group in an arbitrary dimension to be minimally flat (i.e., ‘non-liftable’). These conditions imply very simply stated restrictions on the number of those structural components of the picture that are fixed by the various symmetry operations of the picture. We then also transfer lifting results for symmetric pictures from Euclidean (d − 1)-space to Euclidean d-space via the technique of coning. Finally, we offer some conjectures regarding sufficient conditions for a picture realized generically for a symmetry group to be minimally flat
Non-crossing frameworks with non-crossing reciprocals
We study non-crossing frameworks in the plane for which the classical
reciprocal on the dual graph is also non-crossing. We give a complete
description of the self-stresses on non-crossing frameworks whose reciprocals
are non-crossing, in terms of: the types of faces (only pseudo-triangles and
pseudo-quadrangles are allowed); the sign patterns in the self-stress; and a
geometric condition on the stress vectors at some of the vertices.
As in other recent papers where the interplay of non-crossingness and
rigidity of straight-line plane graphs is studied, pseudo-triangulations show
up as objects of special interest. For example, it is known that all planar
Laman circuits can be embedded as a pseudo-triangulation with one non-pointed
vertex. We show that if such an embedding is sufficiently generic, then the
reciprocal is non-crossing and again a pseudo-triangulation embedding of a
planar Laman circuit. For a singular (i.e., non-generic) pseudo-triangulation
embedding of a planar Laman circuit, the reciprocal is still non-crossing and a
pseudo-triangulation, but its underlying graph may not be a Laman circuit.
Moreover, all the pseudo-triangulations which admit a non-crossing reciprocal
arise as the reciprocals of such, possibly singular, stresses on
pseudo-triangulation embeddings of Laman circuits.
All self-stresses on a planar graph correspond to liftings to piece-wise
linear surfaces in 3-space. We prove characteristic geometric properties of the
lifts of such non-crossing reciprocal pairs.Comment: 32 pages, 23 figure
Panelling planar graphs
The skeleton of a spherical polyhedron may also be viewed as the skeleton of other panelled structures. We characterize those collections of cycles of a planar graph that bound the panels of hinged-panel structures, and distinguish those that arise from spherical polyhedra
The orbit rigidity matrix of a symmetric framework
A number of recent papers have studied when symmetry causes frameworks on a
graph to become infinitesimally flexible, or stressed, and when it has no
impact. A number of other recent papers have studied special classes of
frameworks on generically rigid graphs which are finite mechanisms. Here we
introduce a new tool, the orbit matrix, which connects these two areas and
provides a matrix representation for fully symmetric infinitesimal flexes, and
fully symmetric stresses of symmetric frameworks. The orbit matrix is a true
analog of the standard rigidity matrix for general frameworks, and its analysis
gives important insights into questions about the flexibility and rigidity of
classes of symmetric frameworks, in all dimensions.
With this narrower focus on fully symmetric infinitesimal motions, comes the
power to predict symmetry-preserving finite mechanisms - giving a simplified
analysis which covers a wide range of the known mechanisms, and generalizes the
classes of known mechanisms. This initial exploration of the properties of the
orbit matrix also opens up a number of new questions and possible extensions of
the previous results, including transfer of symmetry based results from
Euclidean space to spherical, hyperbolic, and some other metrics with shared
symmetry groups and underlying projective geometry.Comment: 41 pages, 12 figure
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