On Projective Evolutes of Polygons

IF 0.7 4区数学 Q2 MATHEMATICS

Experimental Mathematics Pub Date : 2022-02-20 DOI:10.1080/10586458.2022.2102095

M. Arnold, R. Schwartz, S. Tabachnikov

引用次数: 0

Abstract

The evolute of a curve is the envelope of its normals. In this note we consider a projectively natural discrete analog of this construction: we deﬁne projective perpendicular bisectors of the sides of a polygon in the projective plane, and study the map that sends a polygon to the new polygon formed by the projective perpendicular bisectors of its sides. We consider this map acting on the moduli space of projective polygons. We analyze the case of pentagons; the moduli space is 2-dimensional in this case. The second iteration of the map has one integral whose level curves are cubic curves, and the transformation on these level curves is conjugated to the map x (cid:55)→ − 4 x mod 1. We also present the results of an experimental study in the case of hexagons.

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关于多边形的投影演化

曲线的渐屈线是其法线的包络线。在本注释中，我们考虑了这种构造的投影自然离散模拟：我们定义了投影平面中多边形边的投影垂直平分线，并研究了将多边形发送到由其边的投影正交平分线形成的新多边形的映射。我们考虑这个映射作用在投影多边形的模空间上。我们分析了五边形的情况；在这种情况下，模量空间是二维的。映射的第二次迭代有一个积分，其水平曲线是三次曲线，并且这些水平曲线上的变换与映射x共轭（cid:55）→ − 4 x mod 1。我们还介绍了六边形情况下的实验研究结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Experimental Mathematics 数学-数学

CiteScore

1.70

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Experimental Mathematics publishes original papers featuring formal results inspired by experimentation, conjectures suggested by experiments, and data supporting significant hypotheses. Experiment has always been, and increasingly is, an important method of mathematical discovery. (Gauss declared that his way of arriving at mathematical truths was "through systematic experimentation.") Yet this tends to be concealed by the tradition of presenting only elegant, fully developed, and rigorous results. Experimental Mathematics was founded in the belief that theory and experiment feed on each other, and that the mathematical community stands to benefit from a more complete exposure to the experimental process. The early sharing of insights increases the possibility that they will lead to theorems: An interesting conjecture is often formulated by a researcher who lacks the techniques to formalize a proof, while those who have the techniques at their fingertips have been looking elsewhere. Even when the person who had the initial insight goes on to find a proof, a discussion of the heuristic process can be of help, or at least of interest, to other researchers. There is value not only in the discovery itself, but also in the road that leads to it.