Rigidity Properties of the Blum Medial Axis

We consider the Blum medial axis of a region in $\mathbb R^n$ with piecewise smooth boundary and examine its "rigidity properties", by which we mean properties preserved under diffeomorphisms of the regions preserving the medial axis. There are several possible versions of rigidity depending on what features of the Blum medial axis we wish to retain. We use a form of the cross ratio from projective geometry to show that in the case of four smooth sheets of the medial axis meeting along a branching submanifold, the cross ratio defines a function on the branching sheet which must be preserved under any diffeomorphism of the medial axis with another. Second, we show in the generic case, along a Y-branching submanifold that there are three cross ratios involving the three limiting tangent planes of the three smooth sheets and each of the hyperplanes defined by one of the radial lines and the tangent space to the Y-branching submanifold at the point, which again must be preserved. Moreover, the triple of cross ratios then locally uniquely determines the angles between the smooth sheets. Third, we observe that for a diffeomorphism of the region preserving the Blum medial axis and the infinitesimal directions of the radial lines, the second derivative of the diffeomorphism at points of the medial axis must satisfy a condition relating the radial shape operators and hence the differential geometry of the boundaries at corresponding boundary points.