Isotopies of generic plane curves
Isotopies of generic plane curves
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通用平面曲线的同位素
DOI：
10.1017/s0017089500005292
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发表时间：
1983
影响因子：
0.5
通讯作者：
John W. Bruce
中科院分区：
文献类型：
--
作者：
John W. Bruce
本文的目的是探讨平面曲线的一般同位素几何的一些方面。我们的主要工具将是Arnol'd [1]关于波前演化的论文。人们可以问的问题是：在平面曲线的一般合痕中，顶点是如何创建和销毁的？二元结构如何演变？高斯地图如何改变？在试图回答这些问题时，我们利用了这样一个事实，即这些现象都自然地与Ak型奇点相关联。这样，Ak+1奇点的分歧集与Ak奇点的判别集重合。因此，我们可以将Arnol'd的结果应用于单参数Legendre奇点族（如奇点），从而得到单参数拉格朗日奇点族（如渐屈线）的信息。对于具有奇异性的函数的分支集，而不是Ak型函数，人们会遇到具有光滑模的问题，见[4]。
The aim of this paper is to explore some facets of the geometry of generic isotopies of plane curves. Our major tool will be the paper of Arnol'd [1] on the evolution of wavefronts. The sort of questions one can ask are: in a generic isotopy of a plane curve how are vertices created and destroyed? How does the dual evolve? How can the Gauss map change? In attempting to answer these questions we are taking advantage of the fact that these phenomena are all naturally associated with singularities of type Ak. Now the bifurcation set of an Ak+1 singularity and the discriminant set of an Ak singularity coincide. So we can apply Arnol'd's results on one parameter families of Legendre (discriminant) singularities (e.g. the duals) to get information on one parameter families of Lagrange (bifurcation) singularities (e.g. the evolutes). For bifurcation sets of functions with singularities other than those of type Ak one runs up against problems with smooth moduli—see [4].