Reshaping Convex Polyhedra
Abstract
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 digontailoring cuts off from P a digon containing v, a subset of P bounded by two equallength 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 Pcutting up Q into pieces pasted nonoverlapping onto P. In the second part of this monograph, we study vertexmerging processes on convex polyhedra (each vertexmerge being in a sense the reverse of a digontailoring), creating embeddings of P into enlarged surfaces. We aim to produce nonoverlapping 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 polynomialtime algorithms.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.03153
 Bibcode:
 2021arXiv210703153O
 Keywords:

 Mathematics  Metric Geometry;
 Computer Science  Computational Geometry;
 52A15;
 52B10;
 52C45;
 53C45;
 68U05 (Primary);
 5202;
 5208;
 52A37;
 52C99 (Secondary);
 F.2.2;
 G.2.2
 EPrint:
 Research monograph. 234 pages, 105 figures, 55 references. arXiv admin note: text overlap with arXiv:2008.01759