脐点顶点集的Perestroikas
Perestroikas of vertex sets at umbilic points
摘要：
将光滑曲面M与平行于M的切面的单参数平面族在点p相交而得到的曲线族下演化的曲线上的所有顶点标记出来。这些顶点通过p跟踪出一个集合，称为M的顶点集。我们将p设为M上的孤立脐点，并描述顶点集在曲面的一般n参数小变形下的perestroikas，以及相应的判别式。我们的结果的一个主要结论是，在某种意义上（见定理2），一般地，研究小n-形变（n大于或等于2）的判别式，简化为研究2-形变的判别式。这项工作的主要动机是计算机视觉和图像分析中形状的中间表示，其中顶点的行为在骨架或Blum中轴曲线的定性变化中起着至关重要的作用。另一方面，曲线顶点的研究引起了极大的兴趣，特别是在几何和奇点理论中，与几个问题，如四顶点定理，曲面的局部几何和焦散线几何。接触几何的发展，特别是V. Arnold关于Legendrian Collapse和Legendrian Sturm Theory的著作，揭示了平面曲线的顶点与Legendrian奇点之间的关系，以及Sturm-Hurwitz定理，使这些经典的学科受到了新的冲击。
英文摘要：
Mark all vertices on a curve evolving under a family of curves obtained by intersecting a smooth surface M with the 1-parameter family of planes parallel to the tangent plane to M at a point p. Those vertices trace out a set, called the vertex set of M through p. We take p to be an isolated umbilic point on M and describe the perestroikas of the vertex set under generic n-parameter small deformations of the surface, as well as the corresponding discriminants. One of the main consequences of our results is that, in some sense (see Theorem 2), generically the study of the discriminants of small n-deformations, with n greater than or equal to 2, simplifies to that of 2-deformations. This work was primarily motivated by the medial representation of shapes in Computer Vision and Image Analysis, where the behaviour of vertices plays a crucial role in the qualitative changes of the skeleton or Blum medial axis of curves. On the other hand, the study of vertices of curves has raised up a great interest in particular in Geometry and Singularity Theory, in line with several problems such as the 4-vertex Theorem, the local geometry of surfaces and Geometry of Caustics. These classical subjects have received a new impulsion due to the development of Contact Geometry, especially in the works of V. Arnold on Legendrian Collapse and Legendrian Sturm Theory showing the relation between vertices of plane curves and Legendrian singularities and Sturm-Hurwitz Theorem.
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