    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.

Authors

08/04/2020

Tailoring for Every Body: Reshaping Convex Polyhedra

Given any two convex polyhedra P and Q, we prove as one of our main resu...
04/13/2020

Reconstructing a Polyhedron between Polygons in Parallel Slices

Given two n-vertex polygons, P=(p_1, …, p_n) lying in the xy-plane at z=...
08/03/2020

Finding Closed Quasigeodesics on Convex Polyhedra

A closed quasigeodesic is a closed loop on the surface of a polyhedron w...
02/05/2018

Un-unzippable Convex Caps

An unzipping of a polyhedron P is a cut-path through its vertices that u...
03/17/2021

We study whether a given graph can be realized as an adjacency graph of ...
05/02/2019

Convex choice, finite choice and sorting

We study the Weihrauch degrees of closed choice for finite sets, closed ...
03/25/2021

Information theoretic parameters of non-commutative graphs and convex corners

We establish a second anti-blocker theorem for non-commutative convex co...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.