淡江大學機構典藏:Item 987654321/121387
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    题名: Construction of a curve by using the state equation of Frenet formula
    作者: J. T. Chen;J. W. Lee;S. K. Kao;Y. T. Chou
    关键词: radius of curvature;Frenet formula;inverse problem;cycloid
    日期: 2021-06-30
    上传时间: 2021-09-25 12:11:10 (UTC+8)
    出版者: Cambridge University Press
    摘要: 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.
    關聯: Journal of Mechanics 37, p.454-465
    DOI: 10.1093/jom/ufab014
    显示于类别:[土木工程學系暨研究所] 期刊論文

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