39 research outputs found
Vertex-Unfoldings of Simplicial Polyhedra
We present two algorithms for unfolding the surface of any polyhedron, all of
whose faces are triangles, to a nonoverlapping, connected planar layout. The
surface is cut only along polyhedron edges. The layout is connected, but it may
have a disconnected interior: the triangles are connected at vertices, but not
necessarily joined along edges.Comment: 10 pages; 7 figures; 8 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
Grid Vertex-Unfolding Orthogonal Polyhedra
An edge-unfolding of a polyhedron is produced by cutting along edges and
flattening the faces to a *net*, a connected planar piece with no overlaps. A
*grid unfolding* allows additional cuts along grid edges induced by coordinate
planes passing through every vertex. A vertex-unfolding permits faces in the
net to be connected at single vertices, not necessarily along edges. We show
that any orthogonal polyhedron of genus zero has a grid vertex-unfolding.
(There are orthogonal polyhedra that cannot be vertex-unfolded, so some type of
"gridding" of the faces is necessary.) For any orthogonal polyhedron P with n
vertices, we describe an algorithm that vertex-unfolds P in O(n^2) time.
Enroute to explaining this algorithm, we present a simpler vertex-unfolding
algorithm that requires a 3 x 1 refinement of the vertex grid.Comment: Original: 12 pages, 8 figures, 11 references. Revised: 22 pages, 16
figures, 12 references. New version is a substantial revision superceding the
preliminary extended abstract that appeared in Lecture Notes in Computer
Science, Volume 3884, Springer, Berlin/Heidelberg, Feb. 2006, pp. 264-27
Branched Coverings, Triangulations, and 3-Manifolds
A canonical branched covering over each sufficiently good simplicial complex
is constructed. Its structure depends on the combinatorial type of the complex.
In this way, each closed orientable 3-manifold arises as a branched covering
over the 3-sphere from some triangulation of S^3. This result is related to a
theorem of Hilden and Montesinos. The branched coverings introduced admit a
rich theory in which the group of projectivities plays a central role.Comment: v2: several changes to the text body; minor correction
Epsilon-Unfolding Orthogonal Polyhedra
An unfolding of a polyhedron is produced by cutting the surface and
flattening to a single, connected, planar piece without overlap (except
possibly at boundary points). It is a long unsolved problem to determine
whether every polyhedron may be unfolded. Here we prove, via an algorithm, that
every orthogonal polyhedron (one whose faces meet at right angles) of genus
zero may be unfolded. Our cuts are not necessarily along edges of the
polyhedron, but they are always parallel to polyhedron edges. For a polyhedron
of n vertices, portions of the unfolding will be rectangular strips which, in
the worst case, may need to be as thin as epsilon = 1/2^{Omega(n)}.Comment: 23 pages, 20 figures, 7 references. Revised version improves language
and figures, updates references, and sharpens the conclusio
Metric combinatorics of convex polyhedra: cut loci and nonoverlapping unfoldings
This paper is a study of the interaction between the combinatorics of
boundaries of convex polytopes in arbitrary dimension and their metric
geometry.
Let S be the boundary of a convex polytope of dimension d+1, or more
generally let S be a `convex polyhedral pseudomanifold'. We prove that S has a
polyhedral nonoverlapping unfolding into R^d, so the metric space S is obtained
from a closed (usually nonconvex) polyhedral ball in R^d by identifying pairs
of boundary faces isometrically. Our existence proof exploits geodesic flow
away from a source point v in S, which is the exponential map to S from the
tangent space at v. We characterize the `cut locus' (the closure of the set of
points in S with more than one shortest path to v) as a polyhedral complex in
terms of Voronoi diagrams on facets. Analyzing infinitesimal expansion of the
wavefront consisting of points at constant distance from v on S produces an
algorithmic method for constructing Voronoi diagrams in each facet, and hence
the unfolding of S. The algorithm, for which we provide pseudocode, solves the
discrete geodesic problem. Its main construction generalizes the source
unfolding for boundaries of 3-polytopes into R^2. We present conjectures
concerning the number of shortest paths on the boundaries of convex polyhedra,
and concerning continuous unfolding of convex polyhedra. We also comment on the
intrinsic non-polynomial complexity of nonconvex polyhedral manifolds.Comment: 47 pages; 21 PostScript (.eps) figures, most in colo
Unfolding and Reconstructing Polyhedra
This thesis covers work on two topics: unfolding polyhedra into the plane and reconstructing polyhedra from partial information. For each topic, we describe previous work in the area and present an array of new research and results.
Our work on unfolding is motivated by the problem of characterizing precisely when overlaps will occur when a polyhedron is cut along edges and unfolded. By contrast to previous work, we begin by classifying overlaps according to a notion of locality. This classification enables us to focus upon particular types of overlaps, and use the results to construct examples of polyhedra with interesting unfolding properties.
The research on unfolding is split into convex and non-convex cases. In the non-convex case, we construct a polyhedron for which every edge unfolding has an overlap, with fewer faces than all previously known examples. We also construct a non-convex polyhedron for which every edge unfolding has a particularly trivial type of overlap. In the convex case, we construct a series of example polyhedra for which every unfolding of various types has an overlap. These examples disprove some existing conjectures regarding algorithms to unfold convex polyhedra without overlaps.
The work on reconstruction is centered around analyzing the computational complexity of a number of reconstruction questions. We consider two classes of reconstruction problems. The first problem is as follows: given a collection of edges in space, determine whether they can be rearranged by translation only to form a polygon or polyhedron. We consider variants of this problem by introducing restrictions like convexity, orthogonality, and non-degeneracy. All of these problems are NP-complete, though some are proved to be only weakly NP-complete. We then consider a second, more classical problem: given a collection of edges in space, determine whether they can be rearranged by translation and/or rotation to form a polygon or polyhedron. This problem is NP-complete for orthogonal polygons, but polynomial algorithms exist for non-orthogonal polygons. For polyhedra, it is shown that if degeneracies are allowed then the problem is NP-hard, but the complexity is still unknown for non-degenerate polyhedra