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 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, and to continuously folding P onto Q. 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.