{"title":"Curves Whose Arcs with a Fixed Startpoint Are Similar","authors":"I. V. Polikanova","doi":"10.3103/s1066369x23110063","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The author has previously put forward a hypothesis that in <i>n-</i>dimensional Euclidean space, curves, any two oriented arcs of which are similar, are rectilinear. The author also proved this statement for dimensions of <i>n</i> = 2 and <i>n</i> = 3. In a space of arbitrary dimension, this hypothesis was confirmed in the class of rectifiable curves. In this study, the author provides a complete solution to this problem that is even stronger: (a) a curve in <i>E</i> <sup><i>n</i></sup>, where any two oriented arcs starting at an common nonfixed point are similar is rectilinear; (b) if a curve in <i>E</i> <sup><i>n</i></sup> has a half-tangent at its boundary point and any two of its oriented arcs emanating from this point are similar, then the curve is rectilinear; (c) if a curve in <i>E</i> <sup><i>n</i></sup> has a tangent at an inner point and all its oriented arcs starting at this point are similar, then the curve is rectilinear. Examples of curves in <i>E</i><sup>2</sup> and <i>E</i><sup>3</sup> are given, that are not rectilinear, although their arcs having a common startpoint are similar, and a complete description of such curves in <i>E</i><sup>2</sup> is given. Research methods are topological and set-theoretic using the apparatus of functional equations.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"62 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Russian Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3103/s1066369x23110063","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The author has previously put forward a hypothesis that in n-dimensional Euclidean space, curves, any two oriented arcs of which are similar, are rectilinear. The author also proved this statement for dimensions of n = 2 and n = 3. In a space of arbitrary dimension, this hypothesis was confirmed in the class of rectifiable curves. In this study, the author provides a complete solution to this problem that is even stronger: (a) a curve in En, where any two oriented arcs starting at an common nonfixed point are similar is rectilinear; (b) if a curve in En has a half-tangent at its boundary point and any two of its oriented arcs emanating from this point are similar, then the curve is rectilinear; (c) if a curve in En has a tangent at an inner point and all its oriented arcs starting at this point are similar, then the curve is rectilinear. Examples of curves in E2 and E3 are given, that are not rectilinear, although their arcs having a common startpoint are similar, and a complete description of such curves in E2 is given. Research methods are topological and set-theoretic using the apparatus of functional equations.
摘要 作者曾提出一个假设，即在 n 维欧几里得空间中，任意两条方向相似的弧都是直线。在任意维度的空间中，这一假设在可直角曲线类中得到了证实。在本研究中，作者为这一问题提供了更强的完整解决方案：(a) E n 中的一条曲线，如果从一个共同的非固定点出发的任意两条定向弧都相似，那么这条曲线就是直角曲线；(b) 如果 E n 中的一条曲线在其边界点有一条半切线，并且从这一点出发的任意两条定向弧都相似，那么这条曲线就是直角曲线；(c) 如果 E n 中的一条曲线在其内点有一条切线，并且从这一点出发的所有定向弧都相似，那么这条曲线就是直角曲线。给出了 E2 和 E3 中曲线的例子，这些曲线虽然具有共同起点的弧相似，但不是直角曲线，并给出了 E2 中此类曲线的完整描述。研究方法是拓扑学和集合论，使用函数方程的工具。
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