Extrapolation methods for approximating arc length and surface area

摘要

A well-known method of estimating the length of a parametric curve in R-d is to sample some points from it and compute the length of the polygon passing through them. In this paper we show that for uniform sampling of regular smooth curves, Richardson extrapolation can be applied repeatedly giving a sequence of derivative-free length estimates of arbitrarily high orders of accuracy. A similar result is derived for the approximation of the area of parametric surfaces.