# Geodesics on a polyhedron

Which sequences of adjacent edges of a polyhedron could be considered to be a geodesic? The edges of a face most surely will not, but the “equator” of the octahedron eventually will. But for what reasons? How do the defining property of a geodesic – having zero geodesic curvature – apply to a sequence of edges?

(One crude guess: any sequence of edges that pairwise don’t share a face? What does this have to do with curvature?)

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It so happens I drew the “equator of a dodecahedron” for one of my papers, so I can’t resist including it here:

Two points I’d like to make. First, a geodesic is a curve that has $\le \pi$ surface to each side at every point. This is Alexandrov’s definition, and it is the right way to think of geodesics on polyhedra. He and Pogorelov called these quasigeodesics (Alexandrov and Zalgaller, Intrinsic Geometry of Surfaces, 1967, p.16; Pogorelov, Extrinsic Geometry of Convex Surfaces, 1973, p.28).

Second, if you allow a doubly covered polygon as a polyhedron of zero volume (as Alexandrov did), then indeed the edges of a face could be a geodesic: consider a doubly covered square, for example. Then the edges of one square face have $\pi$ at all interior-edge points to either side, and $\pi/2$ to either side at the four corners.

Some thoughts are given by Konrad Polthier and Markus Schmies, Straightest geodesics on polyhedral surfaces. Form the abstract:

Geodesic curves are the fundamental concept in geometry to generalize the idea of straight lines to curved surfaces and arbitrary manifolds. On polyhedral surfaces we introduce the notion of discrete geodesic curvature of curves and define straightest geodesics. This allows a unique solution of the initial value problem for geodesics, and therefore a unique movement in a given tangential direction, a property not available in the well-known concept of locally shortest geodesics.

There seems to be an almost trivial definition (which relies on a specific realization of a polyhedron): if the polyhedron is convex and inscribed into a sphere and the central projections of the edges onto the sphere sum up to one great arc or circle, then the edges are geodesic. (Note that each single edge is projected onto a single great arc.)

Due to this definition, the obvious “equators” of the regular octahedron are geodesic, but also the edges of any face of any inscribable polyhedron can be geodesic (in a specific realization). On the other side, some sequences of edges cannot be – in no realization of the polyhedron – geodesic due to this definition, e.g. the zig-zag-sequence around the “equator” of the dodecahedron (?).

It remains open, whether there is a more combinatorial definition of geodesics, say in a polyhedral graph.