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.

我们考虑\（\ mathbb R ^ n \）中具有分段光滑边界的区域的Blum中间轴，并检查其“刚度属性”，这是指在保留中间轴的区域的变态作用下保留的属性。根据我们希望保留的Blum中轴的特征，有几种可能的刚度版本。我们使用射影几何的交叉比率的形式来显示，在四个光滑轴的中间轴沿着分支子流形汇合的情况下，交叉比率定义了分支轴上的一个函数，该函数必须在任何变分的情况下都保留下来。中间轴与另一个。其次，在一般情况下，沿Y-分支子流形，存在三个交叉比率，涉及三个光滑板的三个极限切线平面，以及由其中一条径向线和Y分支子流形的切线空间所定义的每个超平面，这又是必须的被保存。而且，三倍的交叉比然后局部地唯一地确定光滑板之间的角度。第三，我们观察到对于保留Blum中间轴和径向线无穷小方向的区域的亚纯，在亚轴的点上亚纯的二阶导数必须满足与径向形状算符有关的条件，因此，对应边界点处边界的几何形状。