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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.