Locked and Unlocked Smooth Embeddings of Surfaces

Abstract

We study the continuous motion of smooth isometric embeddings of a planar surface in three-dimensional Euclidean space, and two related discrete analogues of these embeddings: polygonal embeddings and flat foldings without interior vertices, under continuous changes of the embedding or folding respectively. For each of these three models of continuous motion, we show that every star-shaped or spiral-shaped domain is unlocked: a continuous motion unfolds it to a flat embedding. Here, a domain is spiral-shaped if it can be reduced to a point by a continuous family of similarities whose images remain within the domain; we prove that polygons with this property can be recognized in linear time. However, we provide an example of a disk with two holes that has locked embeddings: its embeddings are topologically equivalent to a flat embedding but cannot reach a flat embedding by continuous motion.