Extensions of the general polar value based control point specification method in constructing tensor product B-spline surfaces

Abstract

Blossoming has been utilized as a powerful tool for understanding Bezier and B-spline curves. Recently, general polar values (GPV) have been employed as an aid to specify control points for polynomial curves. Based on this technique, it is possible for us to use a set of polar values to control a polynomial curve. In this paper, we first construct linear systems of tensor product B- spline surfaces (TPBS) under the GPV-based framework. We call this extension as the direct extension scheme. Subsequently, based on similar principles, we investigate the issue of extending the linear systems based on degree elevation and partial derivatives evaluation. For the direct extension and degree elevation schemes, high flexibility is provided though a large set of corresponding free parameters. For the partial derivative scheme, partial derivatives on a surface can be used as parameters. In all schemes, a restriction on parameter selection is imposed by the invertibility of the related matrices. Finally, based on the subtle relationships among the basis functions, a fast computation algorithm is developed for evaluating those related basis-function matrices.