Abstract: Rational Bézier surfaces provide an effective tool for geometric design. One aspect of the theory of rational surfaces that is not well understood is what happens when a rational parameterization takes on the value (0/0, 0/0, 0/0) for some parameter value. Such parameter values are called base points of the parameterization. Base points can be introduced into a rational parameterization in Bézier form by setting weights of appropriate control points to zero. By judiciously introducing base points, one can create parameterizations of four-, five- and six-sided surface patches using rational Bézier surfaces defined over triangular domains. Subdivision techniques allow rendering and smooth meshing of such surfaces. Properties of base points also lead to a new understanding of incompatible edge twist methods such as Gregory's patch.