Motivated by a need to define an object-centered reference system determined by the most salient characteristics of the shape, many methods have been proposed, all of which directly or indirectly involve an axis about which the shape is locally symmetric. Recently, a function v, called `the edge strength function', has been successfully used to determine efficiently the axes of local symmetries of 2-d shapes. The level curves of v are interpreted as successively smoother versions of the initial shape boundary. The local minima of the absolute gradient ||▽ν|| along the level curves of ν are shown to be a robust criterion for determining the shape skeleton. More generally, at an extremal point of ||▽ν|| along a level curve, the level curve is locally symmetric with respect to the gradient vector ▽ν. That is, at such a point, the level curve is approximately a conic section whose one of the principal axes coincides with the gradient vector. Thus, the locus of the extremal points of ||▽ν|| along the level curves determines the axes of local symmetries of the shape. In this paper, we extend this method to shapes of arbitrary dimension.