# Straightest Geodesics on Polyhedral Surfaces

**Konrad Polthier and Markus Schmies**

### 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.

An immediate application is the definition of parallel translation of vectors
and a discrete Runge-Kutta method for the integration of vector fields on
polyhedral surfaces. Our definitions only use intrinsic geometric properties of
the polyhedral surface without reference to the underlying

discrete triangulation of the surface or to an ambient space.

**Full Preprint:** Straightest Geodesics on Polyhedral Surfaces (.pdf
1.34 MB)