A Monge-Kantorovich optimal transportation problem between measures supported on the boundaries of domains in R2 is studied with the intent to get an insight into the underlying geometry of a shape recognition problem in computer vision — where one wants to match two simple closed planar curves. The focus is on investigating (i) uniqueness, (ii) smoothness and (iii) geometrical characterization of the solutions. Optimality of these solutions is measured against a cost function defined between the two curves to be compared. Topological constraints allow (iv) a classification of the cost function that strongly dictates the geometry of the optimal solutions.