Tensor-product monotonicity preservation

The preservation of surface shape is important in geometric modelling and approximation theory. In geometric modelling one would like to manipulate the shape of the surface being modelled by the simpler task of manipulating the control points. In scattered data approximation one might wish to approximate bivariate data sampled from a function with a shape property by a spline function sharing that property. Bernstein/Bézier and B-splines surfaces are good candidates for these tasks because they tend to mimic the shapes of their control nets; see Chapters 16 and 17 of [5], [8] and Chapter 6 of [9]. It was shown by Cavaretta and Sharma [4] that tensor-product Bézier surfaces preserve the convexity of their control nets. Later, weaker sufficient conditions for convexity were derived in [6] and [1] and such conditions have been used to approximate scattered data under the constraint of convexity by Jüttler [10]. On the other hand, relatively little seems to be known about monotonicity of tensorproduct Bézier and similar surfaces in directions other than the axial ones. Monotonicity preserving univariate systems have been characterized in [2] and [3] where it was also shown that total positivity (see [11]) is a sufficient condition. This means that there are several systems which preserve monotonicity of curves, for example Bernstein polynomials, B-splines, normalized monomial bases, and β-splines. For surfaces however we might want to demand monotonicity preservation in the two axial directions, axial monotonicity preservation, or in all directions, monotonicity preservation. As we shall see, total positivity is a sufficient condition for axial monotonicity preservation but not for monotonicity preservation. The main result in this paper is to show that tensor-product Bernstein and B-spline bases preserve monotonicity in all directions (see Sections 3 and 4). Moreover the condition that the control net is increasing in a certain direction is equivalent to a set of linear inequalities in the coefficients (see Section 2). Therefore these inequalities provide sufficient linear conditions for the monotonicity of Bézier and B-spline surfaces and could be used as the starting point of a constrained least squares approximation of scattered data.