Locked and unlocked smooth embeddings of surfaces

TL;DR

This paper investigates the continuous deformation of smooth and discrete isometric embeddings of surfaces in three-dimensional space, revealing conditions under which embeddings can be unfolded or are locked.

Contribution

It introduces the concept of locked and unlocked embeddings for surfaces, providing new insights into their deformation properties and topological constraints.

Findings

Star-shaped and spiral-shaped domains are always unlockable to flat embeddings.

Disks with two holes can have locked embeddings that cannot be flattened through continuous motion.

The study bridges smooth and discrete surface embeddings, highlighting topological obstructions.

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. We show that every star-shaped or spiral-shaped domain is unlocked: a continuous motion unfolds it to a flat embedding. However, disks with two holes can have locked embeddings that are topologically equivalent to a flat embedding but cannot reach a flat embedding by continuous motion.