Totally positive bases and progressive iteration approximation

In this paper, we study the progressive iteration approximation property of a curve (tensor product surface) generated by blending a given data point set and a set of basis functions. The curve (tensor product surface) has the progressive iteration approximation property as long as the basis is totally positive and the corresponding collocation matrix is nonsingular. Thus, the B-spline and NURBS curve (surface) have the progressive iteration approximation property, and Bézier curve (surface) also has the property if the corresponding collocation matrix is nonsingular.