  A piecewise flat Finsler metric on a triangulated surface $M$ is a metric
whose restriction to any triangle is a flat triangle in some Minkowski space
with straight edges. One of the main purposes of this work is to study the
properties of geodesics on a piecewise flat Finsler surface, especially when it
meets a vertex. Using the edge-crossing equation, we define two classes of
piecewise flat Finsler surfaces, namely, Landsberg type and Berwald type. We
deduce an explicit condition for a geodesic to be extendable at a vertex, and
define the curvature which measures the \textit{amount} of such extensions. The
dependence of the curvature on an incoming or outgoing tangent direction
corresponds to the feature of flag curvature in Finsler geometry. When the
piecewise flat Finsler surface is of Landsberg type, the curvature is only
relevant to the vertex, and we prove a combinatoric Gauss-Bonnet formula which
generalizes both the Gauss-Bonnet formulas for piecewise flat Riemannian
manifolds and for smooth Landsberg surfaces.
