Source Unfoldings of Convex Polyhedra via Certain Closed Curves

TL;DR

This paper generalizes the source unfolding method for convex polyhedra by using certain closed curves instead of points, enabling a new approach to unfold polyhedral surfaces into simple planar polygons.

Contribution

It introduces a novel unfolding technique based on closed curves that 'live on a cone,' expanding the class of unfoldings beyond points.

Findings

Provides a new method for unfolding convex polyhedra using closed curves.

Ensures non-overlapping unfoldings for a broader class of curves.

Extends the applicability of source unfoldings to include quasigeodesics.

Abstract

We extend the notion of a source unfolding of a convex polyhedron P to be based on a closed polygonal curve Q in a particular class rather than based on a point. The class requires that Q "lives on a cone" to both sides; it includes simple, closed quasigeodesics. Cutting a particular subset of the cut locus of Q (in P) leads to a non-overlapping unfolding of the polyhedron. This gives a new general method to unfold the surface of any convex polyhedron to a simple, planar polygon.