4 research outputs found
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