Convergence analysis for B-spline geometric interpolation

Yunhui Xiong, Guiqing Li, Aihua Mao

Abstract: In this paper, we propose a sufficient condition for the convergence of a geometric algorithm for interpolating a given polygon using non-uniform cubic B-splines. Geometric interpolation uses the given polygon as the initial shape of the control polygon of the B-spline and reduces the approximate error by iteratively updating the control points with the deviations from the corresponding interpolated vertices to their nearest footpoints on the current B-spline curve. The convergence condition is derived by employing a spectral radius estimation technique. The primary goal is to find for each control point a parametric interval within which the nearest footpoint should be confined such that the spectral radius of the error iteration matrix is smaller than 1. A convergent condition for the geometric interpolation of uniform B-splines can be derived as a special case of the new scheme.