This paper provides a curvature-based algorithm to compute locally shortest geodesics on closed triangulated surfaces. These curves, which are called ``constrictions'', are useful for shape segmentation. The key idea of the algorithm is that constrictions are almost plane curves; it first finds well-located simple, plane, closed curves, and then slides them along the surface until a shortest geodesic is reached. An initial curve is defined as a connected component of the intersection between the surface and a plane going through an initial vertex. Initial vertices and planes are determined using approximations of surface curvature.