Changing Tangent and Curvature Data of B-splines via Knot Manipulation

Abstract

Modifications of B-spline knot values change the parametrization and influence the shape of B-spline curves. Via these computations one can modify B-spline data (derivative, curvature value at a curve point, some points of the control polygon, etc.) such that the new parametrization of the curve satisfies special in-put conditions of a B-spline algorithm. We give a detailed analysis of operations on knot vectors determining the parametrization of non-uniform B-spline functions. Different knot manipulation techniques are presented using blossoming approach. We describe a new knot manipulation strategy: repositioning of a knot, which is computed directly without knot insertion and removal. This strategy can be used for clamping the control polygon of B-spline curves. As further applications of the knot manipulation we show two methods which modify the tangent and the curvature data in the starting and end points of B-spline curves. These computations are illustrated with nice examples.