 A little more formally, the best matching procedure works as follows. Consider a class of paths in configuration space, all corresponding to the same path in shape space (i.e., consider a class of paths in configuration space related by the action of the similarity group). As mentioned above, a given metric on configuration space will in general assign to each path in that space a different length; as a result, the length of the associated path in shape space will be underdetermined. However, starting from any given point p in configuration space, one can use the action of the similarity group on configuration space to define a unique curve, by shifting the points of any curve through p along the corresponding orbits of the similarity group (think of these ‘orbits’ as contour lines in configuration space, relating points which differ only by the action of the similarity group) so as to extremize the length assigned to the curve (relative to the metric under consideration). It is this extremized length which is assigned to the unique curve in shape space. With each curve in shape space assigned a unique length, one can then, as usual, specify a principle which selects some such curves as representing nomologically possible worlds, based upon their lengths. (Recall again, for example, that in Lagrangian mechanics it is those curves which extremize an action which are regarded as being dynamically possible.)