Algorithms for generating B-nets and graphically displaying spline surfaces on three- and four-directional meshes

This paper is concerned with the description of an algorithm for generating the B-nets of box splines and an algorithm based on subdivision of these B-nets for graphically displaying spline surfaces on a three-or four-directional mesh. The combination of these two algorithms is shown to compare favorably over the other existing algorithms for this purpose. 0 1 x1. Introduction It is well known that multivariate box spline functions provide a useful approximation family of piecewise polynomials on the uniform grid partition such as the three{ and four{ directional meshes in the bivariate setting (cf. BD83], BH82] for reference on the basic properties of box spline functions.) In some applications, the exact values of a box spline at certain points are also needed. For instance, in bivariate cardinal interpolation (cf. BHR85a], BHR85b]) and quasi-interpolation by using box splines (cf. CD87], CL87a], CDR88]), exact values of the box splines at integers are required. In other applications such as in computer-aided geometric design and computer graphics, one needs graphically displaying a spline surface constructed by translates of box spline B(jX) in the following form