On the length preserving approximation of plane curves by circular arcs (Q1682903)

The paper deals with the problem of approximating of plane curve by two circular arcs. The author investigates and extends the results from \textit{I. Kh. Sabitov} and \textit{A. V. Slovesnov} [Zh. Vychisl. Mat. Mat. Fiz. 50, No. 8, 1347--1356 (2010; Zbl 1224.51031); translation in Comput. Math., Math. Phys. 50, No. 8, 1279--1288 (2010)]. After introducing the basic notation and information about biarcs the author proves that for any convex spiral with an increasing curvature there exists a unique biarcs approximating the spiral of the same length such that the endpoints and tangent vectors at the endpoints of these two curves are identical, and the bounds for the curvatures of the two arcs. Next, the author investigates the inverse problem. He proves the existence of a convex spiral with the same length and the same tangents and boundary curvatures at the points as the initial convex biarc. Moreover, he proves inequalities for the length of a convex spiral arc subject to the given boundary conditions. The paper ends with some computer simulations and conjectures on the curve closeness conditions obtained from these simulations.