Boundary Conditions for the 3-Direction Box-Spline

In their seminal paper Boehm et al[3] show how box splines over regular bivariate grids are defined by coefficients (control points) associated with centres outside the region being defined ("phantom points") as well as with those inside. If we apply the pure subdivision rules derived from the box splines, this means that the configuration shrinks at every step from the original coarse lattice, which includes the phantoms, to the final limit surface. This is inconvenient: we much prefer to have the boundary curve of the limit surface defined as a univariate subdivision curve which is interpolated by the limit surface, and so the practical schemes of importance (Loop[9] and Catmull-Clark[6]) have a 'fix' at the boundary, whereby new vertices associated with the boundary are added at every stage to those defined by the box-spline rules inside the configuration. Unfortunately this leads-in the Catmull-Clark case to the equivalent of the 'natural' end-condition, which 'in spite of its positive sounding name has little to recommend it from an approximation-theoretic point of view' (quoted from de Boor [4]), because the second derivative at the end of each isoline crossing the boundary is zero. In the case of Loop subdivision the second derivative across the edge is always one quarter of that along the edge, and so the limit surface tends to have positive Gaussian curvature at the edge whether or not this is desired. This paper explores the idea that the coefficients of Boehm et al's "phantom points" can be related to those in the domain by use of better boundary conditions. More precisely, we introduce and study extensions of the univariate 'not-a-knot' end-conditions (called 'uniform' by Kershaw in [8]) to 3-direction box-splines, generating boundary conditions whose templates turn out to have simple, elegant and interesting structure. The work is closely related to that of Bejancu and Sabin [2] on the approximation order of semi-cardinal approximation, but is presented here fore a CAGD, rather than approximation theory, audience.