A 'taut string algorithm' for straightening a piecewise linear path in two dimensions

A piecewise linear path in two dimensions is formed by drawing straight lines between adjacent points in the sequence {ui:i=1,2,...,n}. Let Δ be a given positive number such that the length of each straight line segment is at least 3 Δ. We straighten the path in the following way. For i=2,3,...,n-1, we surround ui by a circular ring of radius Δ and centre ui. Then a piece of string that begins at u1 and ends at un is threaded through all the rings in sequence. The new path is constructed by pulling the string tight. An iterative algorithm is proposed that generates the new path to prescribed accuracy. Its convergence is proved and its efficiency is demonstrated by some numerical results.