A Shape Metric for Blum Ribbons

A Blum ribbon is a figure whose boundary is the envelope of a family of circles the centers of which lie along a single unbranched curve called its medial axis. Define a Blum ribbon to be simple if its medial axis is the line segment joining points (0,0) and (1,0). Any Blum ribbon can be made simple by flexing the medial axis, rotating, then dilating, all of which are basic transformations in Blum's geometry of shape. The Lie group SL(2, R) acts on circles centered on the x-axis by linear fractional transformations. By means of this action it is possible to associate to any simple Blum ribbon a curve in SL(2, R). A distance between corresponding points lying on such curves is defined using the bi-invariant metric on SL(2, R). Resulting scale-invariant metrics on the set of figures defined as Blum ribbons are described and it is shown that these metrics can provide effective measures of shape difference.