Turbs { Topologically Unrestricted Rational B-splines

We present a new approach to the construction of piecewise polynomial or rational C k-spline surfaces of arbitrary topological structure. The basic idea is to use exclusively parametric smoothness conditions, and to solve the well known problems at extraordinary points by admitting singular parametrizations. The smoothness of the spline surfaces is guaranteed by specifying a regular smooth re-parametrization explicitly. The resulting space of topologically unrestricted rational B-splines (TURBS) is linear and possesses a natural reenement property. Compared with all known methods the construction principle of TURBS is of striking simplicity and the required polynomial bi-degree is essentially decreased from O(k 2) to d = 2k + 2.