Analytic and Algebraic Properties of Canal Surfaces ?

The envelope of a one-parameter set of spheres with radii r(t) and centers m(t) is a canal surface with m(t) as the spine curve and r(t) as the radii function. This concept is a generalization of the classical notion of an offset of a plane curve. In this paper, we firstly survey the principle geometric features of canal surfaces. In particular, a sufficient condition of canal surfaces without local self-intersection is presented. Moreover,the Gaussian curvature and a simple expression for the area of canal surfaces are given. We also consider the implicit equation f(x, y, z) = 0 of canal surfaces. In particular, the degree of f(x, y, z) is presented. By using the degree of f(x, y, z), a low boundary of the degree of parametrizations representations of canal surfaces is presented. We also prove the low boundary can be reached in some cases.