Curve Interpolation with Constrained Length

摘要

The problem of finding a curve which interpolates at given points such that(approximately) the length of the curve between each two subsequent interpolation points is equal to some given number is considered. Only the functional case is considered. An algorithm which yields an interpolating cubic polynomial spline is given. In case the data is taken from a (smooth enough) function, this spline function converges at least quadratically in the meshsize to the original one. If the mesh is 'regular enough' it is even third order accurate. An extension to the bivariate case is also given. For the univariate case it is shown that the length on each interval of this constructed spline at most differs quadratically in the meshsize from the actual lengths. Assuming regularity on the partition this estimate can also be improved by one order.