曲线在拓扑形变下的准不变量

THE QUASI INVARIANTS OF CURVES UNDER THE TOPOLOGICAL DEFORMATION

平面曲线在拓扑变换作用下的形状差异可以非常巨大． 该文提出曲线的拓扑形变的准不变量概念， 并给出获取这种不变量的方法． 主要想法是寻找新坐标， 使得对于给定的拓扑变换，把原坐标下的拓扑形变转换为新坐标下的平移． 根据Ｌｉｅ变换群对曲线作用的不变性条件， 在积分变换不变的条件下， 通过求解Ｌｉｅ 导数算子导出的偏微分方程求解典则坐标， 从而将原坐标下的拓扑形变转换为典则坐标下的平移． 文中给出了详细的推导和构造不变量的过程， 最后给出了一个数值实例．

The shapes of planar curves distorted under topological transformation are of varying multiplicity. In this paper, a novel concept of quasi invariant under topological deformation of curves is presented and the method of obtaining this sort of quasi invariant is accordingly proposed. The key idea is to find out new coordinates so as to transform from the provided topological distortion on curves in original coordinates to just a shifting translate in the new coordinates. To construct these invariants, the condition of the invariance of Lie transformation group's action on the curves is used, that is, to solve the partial differential equations containing the Lie derivatives under the condition of invariant integral transformation. As a result, when transformed to the canonical coordinates the topologically distortion on curves in the original coodinates is only shifted. The exposition of the derivation and the construction of the invariants are detailed enough. A digital experiment is contained in the last part.