Construction of a curve by using the state equation of Frenet formula
學年 109
學期 2
出版（發表）日期 2021-06-30
作品名稱 Construction of a curve by using the state equation of Frenet formula
作品名稱（其他語言）
著者 J. T. Chen; J. W. Lee; S. K. Kao; Y. T. Chou
單位
出版者
著錄名稱、卷期、頁數 Journal of Mechanics 37, p.454-465
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.
關鍵字 radius of curvature;Frenet formula;inverse problem;cycloid
語言 en
ISSN 1811-8216
期刊性質 國外
收錄於 SCI
產學合作
通訊作者 J. T. Chen
審稿制度 是
國別 GBR
公開徵稿
出版型式 ,電子版
相關連結

機構典藏連結 ( http://tkuir.lib.tku.edu.tw:8080/dspace/handle/987654321/121387 )