Convergence of Geometric Interpolation Using Uniform B-splines

This paper investigates the convergence of an algorithm geometrically interpolating a given polygon using uniform quadratic and cubic B-splines respectively. The geometric interpolation method views the polygon itself as the initial guess of the control polygon of the B-spline and reduces the approximate error by iteratively updating the control points with the deviation from the interpolated vertices to their nearest foot points on the current B-spline curve. We demonstrate that the algorithm usually does not converge if the nearest points are searched on the whole curve and present a sufficient condition under which the algorithm is convergent, for quadratic and cubic B-splines respectively. Furthermore, we introduce a new strategy to update the control points incrementally. Experiments show that though our condition constrains the search range of the nearest points, it can still produce interpolation curves with the same high quality as the original method.