Frobenius可积性，汽车盲点，非倒车镜和全景镜

IF 0.4 4区数学 Q4 MATHEMATICS

American Mathematical Monthly Pub Date : 2023-01-20 DOI:10.1080/00029890.2022.2157659

Elim Hicks, R. A. Hicks, R. Perline, Sarah G. Rody

{"title":"Frobenius可积性，汽车盲点，非倒车镜和全景镜","authors":"Elim Hicks, R. A. Hicks, R. Perline, Sarah G. Rody","doi":"10.1080/00029890.2022.2157659","DOIUrl":null,"url":null,"abstract":"Abstract When an observer looks at a curved mirror, they may sense that a nonlinear map is at work. Here we consider the problem of finding the mirror that realizes a given map. The natural language for such problems is that of planar distributions, and one tool for testing for the existence of solutions is the Frobenius theorem. For situations where exact solutions do not exist, we describe an approximation method that can give good results for applications. Our examples will include non-reversing mirrors, panoramic mirrors, and automotive mirrors without blind spots.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2023-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Frobenius Integrability, Automotive Blind Spots, Non-reversing Mirrors, and Panoramic Mirrors\",\"authors\":\"Elim Hicks, R. A. Hicks, R. Perline, Sarah G. Rody\",\"doi\":\"10.1080/00029890.2022.2157659\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract When an observer looks at a curved mirror, they may sense that a nonlinear map is at work. Here we consider the problem of finding the mirror that realizes a given map. The natural language for such problems is that of planar distributions, and one tool for testing for the existence of solutions is the Frobenius theorem. For situations where exact solutions do not exist, we describe an approximation method that can give good results for applications. Our examples will include non-reversing mirrors, panoramic mirrors, and automotive mirrors without blind spots.\",\"PeriodicalId\":7761,\"journal\":{\"name\":\"American Mathematical Monthly\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2023-01-20\",\"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/10.1080/00029890.2022.2157659\",\"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/10.1080/00029890.2022.2157659","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

当观察者看到一面弯曲的镜子时，他们可能会感觉到一个非线性映射在起作用。这里我们考虑寻找实现给定地图的镜像的问题。这类问题的自然语言是平面分布，而检验解是否存在的一个工具是Frobenius定理。对于不存在精确解的情况，我们描述了一种近似方法，可以给出很好的应用结果。我们的例子将包括非倒车镜、全景镜和无盲点的汽车后视镜。

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

Frobenius Integrability, Automotive Blind Spots, Non-reversing Mirrors, and Panoramic Mirrors

Abstract When an observer looks at a curved mirror, they may sense that a nonlinear map is at work. Here we consider the problem of finding the mirror that realizes a given map. The natural language for such problems is that of planar distributions, and one tool for testing for the existence of solutions is the Frobenius theorem. For situations where exact solutions do not exist, we describe an approximation method that can give good results for applications. Our examples will include non-reversing mirrors, panoramic mirrors, and automotive mirrors without blind spots.

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.