Mesh representation of vasculature is fundamental to many medical applications. The benefit is a clean and tidy appearance in terms of visualization, as well as the possibility of applying computer-assisted intervention and preoperative planning for patients. A vasculature mesh is often reconstructed by iso-surfacing its segmented volume data. Clinicians are usually interested in both the vasculature and its centerline. In this paper, we introduce a mesh centerline extraction approach in the case that volume data are unavailable. The extraction method is inspired by an observation that the vasculature is generally composed of piecewise cylindrical shapes. This observation leads to a conceptually simple but effective strategy to tackle the challenging problem of vasculature centerline extraction, which gracefully combines a branch segmentation scheme and a series of advanced techniques in discrete geometry processing. Our method competes favorably with three state-of-the-art methods in the completeness and accuracy of the extracted centerlines from real human vessels, including the pathological vasculature. Our method also usually leads to maximal and mean extraction errors of less than 1% and 0.5%, respectively.