Abstract: We present an algorithm for computing families of geodesic curves over an open mesh patch to partition the patch into strip-like segments. Specifically, the segments can be well approximated using strips obtained by trimming long, rectangular pieces of material having a prescribed width δ. We call this the width-bounded geodesic strip tiling of a curved surface, a problem with practical applications such as the surfacing of curved roofs. The strips are said to be straight since they are constrained to fit within rectangles of width δ, in contrast to arbitrary, possible highly curved, strip segments. The straightness criterion, as well as a bound on strip widths, distinguishes our problem from ones previously studied for developable surface decomposition. We start with a geodesic curve defined by a user-specified starting point and direction on the input mesh. We then iteratively compute the neighboring geodesics which respect the constraints and lead to optimal use of material (i.e., minimizing the trimmed material) until the surface mesh in question is completely tiled. Our algorithm is exact with respect to the polyhedral geometry of the mesh surface, and it runs on a variety of surfaces with a modest time complexity of O(n^1.5) under reasonable input parameter settings, where n is the mesh size. We also show how the algorithm can be extended by relaxing the constraint that neighboring geodesics span the mesh, thus allowing the algorithm to be applied to meshes with greater undulation.