79 research outputs found
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
A Pseudopolynomial Algorithm for Alexandrov's Theorem
Alexandrov's Theorem states that every metric with the global topology and
local geometry required of a convex polyhedron is in fact the intrinsic metric
of a unique convex polyhedron. Recent work by Bobenko and Izmestiev describes a
differential equation whose solution leads to the polyhedron corresponding to a
given metric. We describe an algorithm based on this differential equation to
compute the polyhedron to arbitrary precision given the metric, and prove a
pseudopolynomial bound on its running time. Along the way, we develop
pseudopolynomial algorithms for computing shortest paths and weighted Delaunay
triangulations on a polyhedral surface, even when the surface edges are not
shortest paths.Comment: 25 pages; new Delaunay triangulation algorithm, minor other changes;
an abbreviated v2 was at WADS 200
Recommended from our members
Discrete Differential Geometry
This is the collection of extended abstracts for the 26 lectures and the open problems session at the second Oberwolfach workshop on Discrete DiïŹerential Geometry
Source Unfoldings of Convex Polyhedra via Certain Closed Curves
Abstract. We extend the notion of a source unfolding of a convex polyhedron P to be based on a closed polygonal curve Q in a particular class rather than based on a point. The class requires that Q âlives on a coneâ to both sides; it includes simple, closed quasigeodesics. Cutting a particular subset of the cut locus of Q (in P) leads to a non-overlapping unfolding of the polyhedron. This gives a new general method to unfold the surface of any convex polyhedron to a simple, planar polygo
Source Unfoldings of Convex Polyhedra via Certain Closed Curves
Abstract. We extend the notion of a source unfolding of a convex polyhedron P to be based on a closed polygonal curve Q in a particular class rather than based on a point. The class requires that Q âlives on a coneâ to both sides; it includes simple, closed quasigeodesics. Cutting a particular subset of the cut locus of Q (in P) leads to a non-overlapping unfolding of the polyhedron. This gives a new general method to unfold the surface of any convex polyhedron to a simple, planar polygo
Polyhedral computational geometry for averaging metric phylogenetic trees
This paper investigates the computational geometry relevant to calculations
of the Frechet mean and variance for probability distributions on the
phylogenetic tree space of Billera, Holmes and Vogtmann, using the theory of
probability measures on spaces of nonpositive curvature developed by Sturm. We
show that the combinatorics of geodesics with a specified fixed endpoint in
tree space are determined by the location of the varying endpoint in a certain
polyhedral subdivision of tree space. The variance function associated to a
finite subset of tree space has a fixed algebraic formula within
each cell of the corresponding subdivision, and is continuously differentiable
in the interior of each orthant of tree space. We use this subdivision to
establish two iterative methods for producing sequences that converge to the
Frechet mean: one based on Sturm's Law of Large Numbers, and another based on
descent algorithms for finding optima of smooth functions on convex polyhedra.
We present properties and biological applications of Frechet means and extend
our main results to more general globally nonpositively curved spaces composed
of Euclidean orthants.Comment: 43 pages, 6 figures; v2: fixed typos, shortened Sections 1 and 5,
added counter example for polyhedrality of vistal subdivision in general
CAT(0) cubical complexes; v1: 43 pages, 5 figure
Reshaping Convex Polyhedra
Given a convex polyhedral surface P, we define a tailoring as excising from P
a simple polygonal domain that contains one vertex v, and whose boundary can be
sutured closed to a new convex polyhedron via Alexandrov's Gluing Theorem. In
particular, a digon-tailoring cuts off from P a digon containing v, a subset of
P bounded by two equal-length geodesic segments that share endpoints, and can
then zip closed.
In the first part of this monograph, we primarily study properties of the
tailoring operation on convex polyhedra. We show that P can be reshaped to any
polyhedral convex surface Q a subset of conv(P) by a sequence of tailorings.
This investigation uncovered previously unexplored topics, including a notion
of unfolding of Q onto P--cutting up Q into pieces pasted non-overlapping onto
P.
In the second part of this monograph, we study vertex-merging processes on
convex polyhedra (each vertex-merge being in a sense the reverse of a
digon-tailoring), creating embeddings of P into enlarged surfaces. We aim to
produce non-overlapping polyhedral and planar unfoldings, which led us to
develop an apparently new theory of convex sets, and of minimal length
enclosing polygons, on convex polyhedra.
All our theorem proofs are constructive, implying polynomial-time algorithms.Comment: Research monograph. 234 pages, 105 figures, 55 references. arXiv
admin note: text overlap with arXiv:2008.0175
Lagrangian Relations and Linear Point Billiards
Motivated by the high-energy limit of the -body problem we construct
non-deterministic billiard process. The billiard table is the complement of a
finite collection of linear subspaces within a Euclidean vector space. A
trajectory is a constant speed polygonal curve with vertices on the subspaces
and change of direction upon hitting a subspace governed by `conservation of
momentum' (mirror reflection). The itinerary of a trajectory is the list of
subspaces it hits, in order. Two basic questions are: (A) Are itineraries
finite? (B) What is the structure of the space of all trajectories having a
fixed itinerary? In a beautiful series of papers Burago-Ferleger-Kononenko
[BFK] answered (A) affirmatively by using non-smooth metric geometry ideas and
the notion of a Hadamard space. We answer (B) by proving that this space of
trajectories is diffeomorphic to a Lagrangian relation on the space of lines in
the Euclidean space. Our methods combine those of BFK with the notion of a
generating family for a Lagrangian relation.Comment: 29 pages, 4 figure
- âŠ