用法涅特公式的状态方程构造曲线

IF 1.5 4区 工程技术 Q3 MECHANICS
J. Chen, J. W. Lee, S. Kao, Y. Chou
{"title":"用法涅特公式的状态方程构造曲线","authors":"J. Chen, J. W. Lee, S. Kao, Y. Chou","doi":"10.1093/jom/ufab014","DOIUrl":null,"url":null,"abstract":"\n In this paper, the available formulae for the curvature of plane curve are reviewed not only for the time-like but also for the space-like parameter curve. Two ways to describe the curve are proposed. One is the straight way to obtain the Frenet formula according to the given curve of parameter form. The other is that we can construct the curve by solving the state equation of Frenet formula subject to the initial position, the initial tangent, normal and binormal vectors, and the given radius of curvature and torsion constant. The remainder theorem of the matrix and the Cayley–Hamilton theorem are both employed to solve the Frenet equation. We review the available formulae of the radius of curvature and examine their equivalence. Through the Frenet formula, the relation among different expressions for the radius of curvature formulae can be linked. Therefore, we can integrate the formulae in the engineering mathematics, calculus, mechanics of materials and dynamics. Besides, biproduct of two new and simpler formulae and the available four formulae in the textbook of the radius of curvature yield the same radius of curvature for the plane curve. Linkage of centrifugal force and radius of curvature is also addressed. A demonstrative example of the cycloid is given. Finally, we use the two new formulae to obtain the radius of curvature for four curves, namely a circle. The equivalence is also proved. Animation for 2D and 3D curves is also provided by using the Mathematica software to demonstrate the validity of the present approach.","PeriodicalId":50136,"journal":{"name":"Journal of Mechanics","volume":" ","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2021-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Construction of a curve by using the state equation of Frenet formula\",\"authors\":\"J. Chen, J. W. Lee, S. Kao, Y. Chou\",\"doi\":\"10.1093/jom/ufab014\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"\\n In this paper, the available formulae for the curvature of plane curve are reviewed not only for the time-like but also for the space-like parameter curve. Two ways to describe the curve are proposed. One is the straight way to obtain the Frenet formula according to the given curve of parameter form. The other is that we can construct the curve by solving the state equation of Frenet formula subject to the initial position, the initial tangent, normal and binormal vectors, and the given radius of curvature and torsion constant. The remainder theorem of the matrix and the Cayley–Hamilton theorem are both employed to solve the Frenet equation. We review the available formulae of the radius of curvature and examine their equivalence. Through the Frenet formula, the relation among different expressions for the radius of curvature formulae can be linked. Therefore, we can integrate the formulae in the engineering mathematics, calculus, mechanics of materials and dynamics. Besides, biproduct of two new and simpler formulae and the available four formulae in the textbook of the radius of curvature yield the same radius of curvature for the plane curve. Linkage of centrifugal force and radius of curvature is also addressed. A demonstrative example of the cycloid is given. Finally, we use the two new formulae to obtain the radius of curvature for four curves, namely a circle. The equivalence is also proved. Animation for 2D and 3D curves is also provided by using the Mathematica software to demonstrate the validity of the present approach.\",\"PeriodicalId\":50136,\"journal\":{\"name\":\"Journal of Mechanics\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":1.5000,\"publicationDate\":\"2021-06-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Mechanics\",\"FirstCategoryId\":\"5\",\"ListUrlMain\":\"https://doi.org/10.1093/jom/ufab014\",\"RegionNum\":4,\"RegionCategory\":\"工程技术\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MECHANICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mechanics","FirstCategoryId":"5","ListUrlMain":"https://doi.org/10.1093/jom/ufab014","RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MECHANICS","Score":null,"Total":0}
引用次数: 1

摘要

本文不仅对类时间参数曲线,而且对类空间参数曲线,都给出了平面曲线曲率的可用公式。提出了两种描述曲线的方法。一种是根据给定的参数形式的曲线得到Frenet公式的直接方法。另一种是,我们可以通过求解Frenet公式中受初始位置、初始切线、法线和二法线矢量以及给定曲率半径和扭转常数约束的状态方程来构造曲线。矩阵的余数定理和Cayley-Hamilton定理都用于求解Frenet方程。我们回顾了现有的曲率半径公式,并检验了它们的等价性。通过Frenet公式,可以将曲率半径公式的不同表达式之间的关系联系起来。因此,我们可以把工程数学、微积分、材料力学和动力学中的公式综合起来。此外,两个新的更简单的曲率半径公式和教科书中现有的四个曲率半径公式的双乘积产生了相同的平面曲线曲率半径。还讨论了离心力与曲率半径之间的联系。给出了摆线的一个实例。最后,我们使用这两个新公式来获得四条曲线的曲率半径,即一个圆。并证明了其等价性。还通过使用Mathematica软件提供2D和3D曲线的动画来证明本方法的有效性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Construction of a curve by using the state equation of Frenet formula
In this paper, the available formulae for the curvature of plane curve are reviewed not only for the time-like but also for the space-like parameter curve. Two ways to describe the curve are proposed. One is the straight way to obtain the Frenet formula according to the given curve of parameter form. The other is that we can construct the curve by solving the state equation of Frenet formula subject to the initial position, the initial tangent, normal and binormal vectors, and the given radius of curvature and torsion constant. The remainder theorem of the matrix and the Cayley–Hamilton theorem are both employed to solve the Frenet equation. We review the available formulae of the radius of curvature and examine their equivalence. Through the Frenet formula, the relation among different expressions for the radius of curvature formulae can be linked. Therefore, we can integrate the formulae in the engineering mathematics, calculus, mechanics of materials and dynamics. Besides, biproduct of two new and simpler formulae and the available four formulae in the textbook of the radius of curvature yield the same radius of curvature for the plane curve. Linkage of centrifugal force and radius of curvature is also addressed. A demonstrative example of the cycloid is given. Finally, we use the two new formulae to obtain the radius of curvature for four curves, namely a circle. The equivalence is also proved. Animation for 2D and 3D curves is also provided by using the Mathematica software to demonstrate the validity of the present approach.
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
Journal of Mechanics
Journal of Mechanics 物理-力学
CiteScore
3.20
自引率
11.80%
发文量
20
审稿时长
6 months
期刊介绍: The objective of the Journal of Mechanics is to provide an international forum to foster exchange of ideas among mechanics communities in different parts of world. The Journal of Mechanics publishes original research in all fields of theoretical and applied mechanics. The Journal especially welcomes papers that are related to recent technological advances. The contributions, which may be analytical, experimental or numerical, should be of significance to the progress of mechanics. Papers which are merely illustrations of established principles and procedures will generally not be accepted. Reports that are of technical interest are published as short articles. Review articles are published only by invitation.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信