The continuous non-linear approximation of procedurally defined curves using integral B-splines

Abstract

This paper outlines an algorithm for the continuous non-linear approximation of procedurally defined curves. Unlike conventional approximation methods using the discrete L_2 form metric with sampling points, this algorithm uses the continuous L_2 form metric based on minimizing the integral of the least square error metric between the original and approximate curves. Expressions for the optimality criteria are derived based on exact B-spline integration. Although numerical integration may be necessary for some complicated curves, the use of numerical integration is minimized by a priori explicit evaluations. Plane or space curves with high curvatures and/or discontinuities can also be handled by means of an adaptive knot placement strategy. It has been found that the proposed scheme is more efficient and accurate compared to currently existing interpolation and approximation methods.