Analytic and algebraic properties of canal surfaces

The envelope of a one-parameter set of spheres with radii r(t)r(t) and centers m(t)m(t) is a canal surface with m(t)m(t) as the spine curve and r(t)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 for canal surfaces without local self-intersection is presented. Moreover, a simple expression for the area and Gaussian curvature of canal surfaces are given. We also consider the implicit equation f(x,y,z)=0f(x,y,z)=0 of canal surfaces with the degree of f(x,y,z)f(x,y,z) presented. By using the degree of f(x,y,z)f(x,y,z), a low boundary of the degree of parameterizations representations of canal surfaces is presented.