Is the radial distance really a distance? An analysis of its properties and interest for the matching of polygon features

In this paper, we examine the properties of the radial distance which has been used as a tool to compare the shape of simple surfacic objects. We give a rigorous definition of the radial distance and derive its theoretical properties, and in particular under which conditions it satisfies the distance properties. We show how the computation of the radial distance can be implemented in practice and made faster by the use of an analytical formula and a Fast Fourier Transform. Finally, we conduct experiments to measure how the radial distance is impacted by perturbation and generalization and we give abacuses and thresholds to deduce when buildings are likely to be homologous or non-homologous given their radial distance.