6 research outputs found
Conical Existence of Closed Curves on Convex Polyhedra
Let C be a simple, closed, directed curve on the surface of a convex
polyhedron P. We identify several classes of curves C that "live on a cone," in
the sense that C and a neighborhood to one side may be isometrically embedded
on the surface of a cone Lambda, with the apex a of Lambda enclosed inside (the
image of) C; we also prove that each point of C is "visible to" a. In
particular, we obtain that these curves have non-self-intersecting developments
in the plane. Moreover, the curves we identify that live on cones to both sides
support a new type of "source unfolding" of the entire surface of P to one
non-overlapping piece, as reported in a companion paper.Comment: 24 pages, 15 figures, 6 references. Version 2 includes a solution to
one of the open problems posed in Version 1, concerning quasigeodesic loop
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
The Geometry Conference
This volume presents easy-to-understand yet surprising properties obtained using topological, geometric and graph theoretic tools in the areas covered by the Geometry Conference that took place in Mulhouse, France from September 7–11, 2014 in honour of Tudor Zamfirescu on the occasion of his 70th anniversary. The contributions address subjects in convexity and discrete geometry, in distance geometry or with geometrical flavor in combinatorics, graph theory or non-linear analysis. Written by top experts, these papers highlight the close connections between these fields, as well as ties to other domains of geometry and their reciprocal influence. They offer an overview on recent developments in geometry and its border with discrete mathematics, and provide answers to several open questions. The volume addresses a large audience in mathematics, including researchers and graduate students interested in geometry and geometrical problems