552 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
Flip Distance Between Triangulations of a Planar Point Set is APX-Hard
In this work we consider triangulations of point sets in the Euclidean plane,
i.e., maximal straight-line crossing-free graphs on a finite set of points.
Given a triangulation of a point set, an edge flip is the operation of removing
one edge and adding another one, such that the resulting graph is again a
triangulation. Flips are a major way of locally transforming triangular meshes.
We show that, given a point set in the Euclidean plane and two
triangulations and of , it is an APX-hard problem to minimize
the number of edge flips to transform to .Comment: A previous version only showed NP-completeness of the corresponding
decision problem. The current version is the one of the accepted manuscrip
Reconstruction of Orthogonal Polyhedra
In this thesis I study reconstruction of orthogonal polyhedral surfaces
and orthogonal polyhedra from partial information about their
boundaries. There are three main questions for which I provide novel
results. The first question is "Given the dual graph, facial angles and
edge lengths of an orthogonal polyhedral surface or polyhedron, is it
possible to reconstruct the dihedral angles?" The second question is
"Given the dual graph, dihedral angles and edge lengths of an
orthogonal polyhedral surface or polyhedron, is it possible to
reconstruct the facial angles?" The third question is "Given the
vertex coordinates of an orthogonal polyhedral surface or polyhedron, is
it possible to reconstruct the edges and faces, possibly after
rotating?"
For the first two questions, I show that the answer is "yes" for
genus-0 orthogonal polyhedra and polyhedral surfaces under some
restrictions, and provide linear time algorithms. For the third
question, I provide results and algorithms for orthogonally convex
polyhedra. Many related problems are studied as well
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