Ambient Approximation on Hypersurfaces

We present a novel general approach for solving approximation problems on hypersurfaces drawing on standard approximation techniques in the ambient space. The method is capable of transferring approximation orders from the ambient space to the hypersurface under mild conditions. When based on tensor product B-splines of order n and knot spacing h, the resulting approximant is \(C^{n-2}\), while the error decays at the optimal rate \(h^n\). Further, the method is easy to implement and, when applying quasi interpolation techniques, optimally efficient as the computational expense is linear in the number of spline coefficients. Applications include the representation of smooth surfaces of arbitrary genus.