Among the various existing and mathematically equivalent definitions of the skeleton, we consider the set of critical points of the Euclidean distance transform of the shape. The problem of detecting these points and using them to generate a skeleton that is stable , thin and homotopic to the shape has been the focus of numerous papers. Skeleton branches correspond to ridges of the distance map, i.e. continuous lines of points that are local maxima of the distance in at least one direction. Extracting these ridges is a non-trivial task on a discrete grid. In this context, the average outward flux, used in the Hamilton-Jacobi skeleton [43], and the ridgeness measure [28] have been proposed as ridge detectors. We establish the mathematical relation between these detectors and, extending the work in [18], we study various local shape configurations, on which closed-form expressions or approximations of the average outward flux and ridgeness can be derived. In addition , we conduct experiments to assess the accuracy of skeletons generated using these measures, and study the influence of their respective parameters.