The Parametrization of Canal Surfaces and the Decomposition of Polynomials into a Sum of Two Squares

A canal surface in R3, generated by a parametrizedcurveC = m(t), is the Zariski closure of the envelope of the set of spheres with radius r(t) centered at m(t). This concept is a generalization of the classical notion of an offsets of a plane curve: first, the canal surface is a surface in 3-space rather than a curve inR2 and second, the radius function r(t) is allowed to vary with the parametert . In case r(t) = const, the resulting envelope is called a pipe surface. In this paper we develop an elementary symbolic method for generating rational parametrizations of canal surfaces generated by rational curves m(t) with rational radius variation r(t). This method leads to the problem of decomposing a polynomial into a sum of two squares over R. We discuss decomposition algorithms which give symbolic and numerical answers to this problem.