论文标题
蓝光内侧轴的刚度特性
Rigidity Properties of the Blum Medial Axis
论文作者
论文摘要
我们考虑了带有分段平滑边界的$ \ mathbb r^n $中一个区域的Blum内侧轴，并检查其“刚度特性”，我们的意思是在保留内侧轴的区域的差异性下保存的特性。刚性有几种可能的刚度，具体取决于我们希望保留的Blum内侧轴的特征。我们使用一种从射影几何形状的交叉比例形式表明，在沿分支亚策略沿着内侧轴的四个光滑的轴相交的情况下，交叉比定义了在分支纸上的功能，必须在内侧轴的任何差异下保留，而该功能必须与另一个相同。其次，在通用情况下，沿着Y分支的子手势显示，有三个跨比涉及三个平滑纸的三个极限切线，以及每个由径向线之一定义的超平面，而由Y Branching Submanifold定义的每个超级平面，必须再次保留。此外，交叉比率的三倍在局部唯一地决定了光滑床单之间的角度。第三，我们观察到，对于保存蓝光内侧轴的区域的差异和径向线的无限方向，径向线的无限方向，内侧轴点处的差异性的第二个衍生物必须满足与径向形状算子相关的条件，从而使边界的差异几何相关。
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.
