Guided spline surfaces

We separate the conceptual design and the representation of high quality surfaces by prescribing local shape via guide surfaces and then sampling these guides with a finite number of tensor-product patches. The paper develops a family of algorithms that allow trading polynomial degree for smoothness near the extraordinary points where more or fewer than four tensor-product patches meet. A key contribution are rules for a capping of a multi-sided hole by a small number of polynomial patches. The construction of highest quality creates first a G1 cap of patches of degree (6,6) and then perturbs it to yield an exact G2 cap of degree (8,8). Since this perturbation is so small that its effect is typically not perceptible even in curvature display, the unperturbed surface of degree (6,6) is an excellent alternative. Reducing the degree of the rings to (5,5), respectively (4,4), by choice of a different parameterization, increases the number of G1 transition curves within the cap but does not alter the shape appreciably.