 Informally, the goal of best matching is to use the similarity group to minimize the intrinsic difference between successive points along a history in configuration space. To take a simple example drawn from Barbour (1999), consider the history of a particular triangle: the triangle may, along that history, rotate, dilate, alter its internal angles, and so on. However, at each point best matching allows one to act on the triangle with similarity transformations; thereby, triangles which at successive points along a history differ merely by a translation, rotation or dilatation will be regarded as being identical after best matching. In this way, a ‘best matched’ history is selected, in which the intrinsic differences between successive states of the system under consideration (in the above example, the triangle) are minimised. While a metric on configuration space will in general assign a different length to histories differing by the action of the similarity group, the length of the best matched history, constructed via the above procedure, will induce a unique length of paths, and therefore metric, on shape space.