INTRODUCTION

Both shortest paths and geodesics are paths on a surface which “unfold” or “develop” on a plane to straight lines. This gives them a special role in unfolding, and underlies the two known (general) unfoldings of a convex polyhedron to a nonoverlapping polygon, the source and the star unfolding (mentioned previously on p. 306). In this chapter we describe these two unfoldings, as well as investigate shortest paths and geodesics for their own sake and for their possible future application to unfolding.

Let P be a polyhedral surface embedded in ℝ3, composed of flat faces. We will insist that each vertex be a “true,” nonflat vertex, one with incident face angle ≠ 2π. A shortest path on P between two points x and y on P is a curve connecting x and y whose length, measured on the surface, is shortest among all curves connecting the points on P. Several important properties of shortest paths are as follows:

1. There always exists a shortest path between any two points, but it may not be unique: several distinct but equally shortest curves may connect the points.

2. Shortest paths are simple in that they never self-cross (otherwise they could be shortcut).

3. […]