C2 Surfaces Built from Zero Sets of the 7-Direction Box Spline

Curvature continuous surfaces can be generated by tracing the zero set of a trivariate box spline of degree four. This box spline is deened by seven directions that form a regular partition of space into tetrahedra so that an eecient approximate evaluation is obtained via subdivision and piecewise linear approximation. There is a natural representation of hierarchical data by perturbation at various levels of subdivision. The approach is contrasted with curvature continuous parametric splines. x1. Introduction Parametric curvature continuous surfaces following the outlines of a regular , checkerboard structured mesh are in common use in the form of bicubic tensor product splines. Accomodating a non checkerboard layout is far more diicult, because the problem is truly bivariate in nature. Only a few C 2 surface constructions for irregular layouts are currently known ((Seidel], Ha-gen,Pottmann], Hahn], Peters]). None of these representations is completely satisfactory, placing restrictions on the input data, are slow to evaluate, lack guaranteed shape properties or are based on lenghty and hence not elegant formulas. In contrast, it is straightforward to deene a curvature continuous surface as the zero set of a trivariate, piecewise polynomial, C 2 function. The zero set of such a function is of the same smoothness class since the expansions of the derivatives agree in all directions. By choosing the coeecients of this function suitably, we may hope to capture the geometry and allow for an approximate parametrization of the surface. There are currently two main approaches to deening the individual pieces of a function whose zero set represents a surface. The rst is to locally generate a shell-like structure consisting of polygonal cells that follow the outlines of and enclose the intended surface. This approach is illustrated by the recent work in Sederberg], Dahmen], Guo], Dahmen, Thamm-Schaar], Bajaj] and Middleditch]. Creating the appropriate shell structure can be challenging and leads to special cases, say in the case of coplanar pieces. The alternative approach deenes a function on a regular,