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 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 ) = 0 of canal surfaces with the degree of f ( x , y , z ) presented. By using the degree of f ( x , y , z ) , a low boundary of the degree of parameterizations representations of canal surfaces is presented.