用反射论焦散的顶点:关于四顶点定理和雅可比最后几何陈述的台球变型

IF 0.4 4区数学 Q4 MATHEMATICS

American Mathematical Monthly Pub Date : 2023-03-17 DOI:10.1080/00029890.2023.2179842

Gil Bor, S. Tabachnikov

{"title":"用反射论焦散的顶点:关于四顶点定理和雅可比最后几何陈述的台球变型","authors":"Gil Bor, S. Tabachnikov","doi":"10.1080/00029890.2023.2179842","DOIUrl":null,"url":null,"abstract":"Abstract A point source of light is placed inside an oval. The nth caustic by reflection is the envelope of the light rays emanating from the light source after n reflections off the curve. We show that, for a generic point light source, each of these caustics has at least 4 cusps. This is a billiard variation on Jacobi’s Last Geometric Statement concerning the number of cusps of the conjugate locus of a point on a convex surface. We present various proofs, using different ideas, including the curve shortening flow and Legendrian knot theory.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":"130 1","pages":"454 - 467"},"PeriodicalIF":0.4000,"publicationDate":"2023-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Cusps of Caustics by Reflection: Billiard Variations on the Four Vertex Theorem and on Jacobi’s Last Geometric Statement\",\"authors\":\"Gil Bor, S. Tabachnikov\",\"doi\":\"10.1080/00029890.2023.2179842\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract A point source of light is placed inside an oval. The nth caustic by reflection is the envelope of the light rays emanating from the light source after n reflections off the curve. We show that, for a generic point light source, each of these caustics has at least 4 cusps. This is a billiard variation on Jacobi’s Last Geometric Statement concerning the number of cusps of the conjugate locus of a point on a convex surface. We present various proofs, using different ideas, including the curve shortening flow and Legendrian knot theory.\",\"PeriodicalId\":7761,\"journal\":{\"name\":\"American Mathematical Monthly\",\"volume\":\"130 1\",\"pages\":\"454 - 467\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2023-03-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"American Mathematical Monthly\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1080/00029890.2023.2179842\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"American Mathematical Monthly","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1080/00029890.2023.2179842","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

摘要一个点光源被放置在一个椭圆内。反射的第n个焦散是在从曲线反射n次之后从光源发出的光线的包络。我们表明，对于通用点光源，这些焦散中的每一个都至少有4个尖端。这是关于凸表面上点的共轭轨迹的尖端数的Jacobi最后一个几何陈述的台球变体。我们用不同的思想提出了各种各样的证明，包括曲线缩短流和勒让德结理论。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

On Cusps of Caustics by Reflection: Billiard Variations on the Four Vertex Theorem and on Jacobi’s Last Geometric Statement

Abstract A point source of light is placed inside an oval. The nth caustic by reflection is the envelope of the light rays emanating from the light source after n reflections off the curve. We show that, for a generic point light source, each of these caustics has at least 4 cusps. This is a billiard variation on Jacobi’s Last Geometric Statement concerning the number of cusps of the conjugate locus of a point on a convex surface. We present various proofs, using different ideas, including the curve shortening flow and Legendrian knot theory.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

American Mathematical Monthly Mathematics-General Mathematics

CiteScore

0.80

自引率

20.00%

发文量

127

审稿时长

6-12 weeks

期刊介绍： The Monthly''s readers expect a high standard of exposition; they look for articles that inform, stimulate, challenge, enlighten, and even entertain. Monthly articles are meant to be read, enjoyed, and discussed, rather than just archived. Articles may be expositions of old or new results, historical or biographical essays, speculations or definitive treatments, broad developments, or explorations of a single application. Novelty and generality are far less important than clarity of exposition and broad appeal. Appropriate figures, diagrams, and photographs are encouraged. Notes are short, sharply focused, and possibly informal. They are often gems that provide a new proof of an old theorem, a novel presentation of a familiar theme, or a lively discussion of a single issue. Abstracts for articles or notes should entice the prospective reader into exploring the subject of the paper and should make it clear to the reader why this paper is interesting and important. The abstract should highlight the concepts of the paper rather than summarize the mechanics. The abstract is the first impression of the paper, not a technical summary of the paper. Excessive use of notation is discouraged as it can limit the interest of the broad readership of the MAA, and can limit search-ability of the article.