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Moving Frames and Differential Invariants on Fully Affine Planar Curves
Bulletin of the Malaysian Mathematical Sciences Society ( IF 1.2 ) Pub Date : 2019-12-04 , DOI: 10.1007/s40840-019-00864-z
Yun Yang , Yanhua Yu

In this paper, by the affine analogue of the fundamental theorem for Euclidean planar curves, we classify the affine curves with constant affine curvatures. Note that we use the fully affine group and not the equi-affine subgroup consisting of area-preserving affine transformations. (Caution: much of the literature omits the “equi-” in their treatment.) According to the equivariant method of moving frames, explicit formulas for the generating affine differential invariants and invariant differential operators are constructed. At the same time, by using the fact that the affine transformation group GA\((2,\mathbb {R})\) can factor as a product of two subgroup \(B\cdot \mathrm{SE}(2,\mathbb {R})\) and the moving frame of the subgroup SE\((2,\mathbb {R})\), we build the moving frame of GA\((2,\mathbb {R})\) and obtain the relations among invariants of group GA\((2,\mathbb {R})\) and its subgroup SE\((2,\mathbb {R})\). Applying the affine curvature to recognize affine equivalent objects is considered in the last part of this paper.

中文翻译:

完全仿射平面曲线上的运动框架和微分不变量

在本文中,通过欧几里得平面曲线基本定理的仿射类比,对具有恒定仿射曲率的仿射曲线进行分类。请注意,我们使用完全仿射组,而不是由保留面积的仿射变换组成的等价仿射子组。(警告:很多文献都忽略了“等式”。)根据运动框架的等变方法,构造了用于生成仿射微分不变量和不变微分算子的显式。同时,通过使用仿射变换组GA \((2,\ mathbb {R})\)可以分解为两个子组\(B \ cdot \ mathrm {SE}(2,\ mathbb {R})\)和子组SE \((2,\ mathbb {R})\)的移动框架,我们建立GA \((2,\ mathbb {R})\)的移动框架,并获得GA \((2,\ mathbb {R})\)组及其子组SE \(( 2,\ mathbb {R})\)。本文的最后一部分考虑了使用仿射曲率识别仿射等效对象。
更新日期:2019-12-04
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