A symbolic-numerical method for computing approximate parameterizations of canal surfaces
A canal surface is the envelope of a one-parameter family of spheres centered at the spine curve m ( t ) and with the radii described by the function r(t) r ( t ) . It was proved in Peternell and Pottmann (1997)[9] that any canal surface to a rational spine curve and a rational radius function possesses a rational parameterization. Then a symbolic method for generating rational parameterizations of canal surfaces was developed in Landsmann etal. (2001)[21]. Indeed, this method leads to the problem of decomposing a polynomial into a sum of two squares over reals, which is solved numerically in general. Hence, approximate techniques generating a parameterization within a certain region of interest are also worth studying. In this paper, we present a method for the computation of approximate rational parameterizations of canal surfaces. A main feature of our approach is a combination of symbolic and numerical techniques yielding approximate topology-based parameterizations of contour curves which are then applied to compute an approximate parameterization of the given canal surface. The algorithm is mainly suitable for implicit blend surfaces of the canal-surface-type.
approximate parameterization canal surface rational parameterization implicit blends critical points