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

Abstract

A canal surface in R 3 , generated by a parametrized curve C = 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 in R 2 and second, the radius function r(t) is allowed to vary with the parameter t. 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.

Rational Offset Surfaces and their Modeling Applications

“…The related problem of finding a decomposition of a real polynomial as a sum of two squares over Q was considered in [25]. It was proved that the problem is equivalent to partial factorization of of the polynomial, and a decomposition algorithm was presented in case the solution is defined over Q.…”

Recognizing implicitly given rational canal surfaces

“…If it is not possible to reconstruct M 2 (see the details in Section 3), or the canal surface Y corresponding to the computed M 2 differs from X then we can state that the given X is not a rational canal surface. Otherwise the output of the algorithm will be positive and M 2 may be then used to parameterize the surface, see [2]. Lemma 2.2.…”

“…It was proved in [1] that any canal surface with a rational spine curve (a set of all centers of moving spheres) and a rational radius function possesses a rational parameterization. An algorithm for generating rational parameterizations of canal surfaces was developed and investigated in [2]. The class of rational canal surfaces with a rational spine curve and a rational radius function is a proper subset of the class of rational canal surfaces -all canal surfaces with a rational spine curve and rational squared radius function admit a rational parameterization, cf.…”

On Characterization Canal Surfaces around Timelike Horizontal Biharmonic Curves in Lorentzian Heisenberg Group Heis3

“…The aim of this paper is to study a canal surface surrounding a timelike horizontal biharmonic curve in the Lorentzian Heisenberg group Heis 3 .…”