Cusps of caustics by reflection in ellipses

IF 1 2区数学 Q1 MATHEMATICS

Journal of the London Mathematical Society-Second Series Pub Date : 2024-11-19 DOI:10.1112/jlms.70033

Gil Bor, Mark Spivakovsky, Serge Tabachnikov

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引用次数: 0

Abstract

This paper is concerned with the billiard version of Jacobi's last geometric statement and its generalizations. Given a non-focal point $O$ inside an elliptic billiard table, one considers the family of rays emanating from $O$ and the caustic $Γ_{n}$ of the reflected family after $n$ reflections off the ellipse, for each positive integer $n$ . It is known that $Γ_{n}$ has at least four cusps and it has been conjectured that it has exactly four (ordinary) cusps. The present paper presents a proof of this conjecture in the special case when the ellipse is a circle. In the case of an arbitrary ellipse, we give an explicit description of the location of four of the cusps of $Γ_{n}$ , though we do not prove that these are the only cusps.

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通过椭圆中的反射实现凹面的顶点

本文关注的是雅可比最后一个几何陈述的台球桌版本及其一般化。给定一个椭圆台球桌内的非焦点 O $O$，考虑从 O $O$ 射出的射线族，以及在每个正整数 n $n$ 反射出椭圆 n $n$ 后反射族的苛值 Γ n $ \Gamma _n$ 。众所周知，Γ n $\Gamma _n$至少有四个尖顶，而且有人猜想它正好有四个（普通）尖顶。本文在椭圆是圆的特殊情况下证明了这一猜想。在任意椭圆的情况下，我们明确描述了 Γ n $\Gamma _n$ 的四个尖顶的位置，尽管我们并没有证明这些尖顶是唯一的尖顶。

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

Journal of the London Mathematical Society-Second Series 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

186

审稿时长

6-12 weeks

期刊介绍： The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.