629 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
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational
geometry is presented.Comment: 7 pages; 72 reference
Ununfoldable Polyhedra with Convex Faces
Unfolding a convex polyhedron into a simple planar polygon is a well-studied
problem. In this paper, we study the limits of unfoldability by studying
nonconvex polyhedra with the same combinatorial structure as convex polyhedra.
In particular, we give two examples of polyhedra, one with 24 convex faces and
one with 36 triangular faces, that cannot be unfolded by cutting along edges.
We further show that such a polyhedron can indeed be unfolded if cuts are
allowed to cross faces. Finally, we prove that ``open'' polyhedra with
triangular faces may not be unfoldable no matter how they are cut.Comment: 14 pages, 9 figures, LaTeX 2e. To appear in Computational Geometry:
Theory and Applications. Major revision with two new authors, solving the
open problem about triangular face
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