Paper Info
Title
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. References References 1 J. Bochnak M. Coste M.-F. Roy Géometrie Algébrique Réelle 1987 Springer Berlin 2 D. Hilbert Mathematische probleme Arch. Math. Phys. 1 1901 213 237 3 E. Hillgarter, G. Landsmann, J. Schicho, F. Winkler 4 E. Hillgarter F. Winkler Points on algebraic curves and the parametrization problem D. Wang H.Shi L. Fariñas Automated Deduction in Geometry 1997 Springer 185 203 5 S. Landau Factoring polynomials over algebraic number fields SIAM J. Comput. 14 1985 184 195 6 G. Landsmann J. Schicho F. Winkler E. Hillgarter C. Traverso Symbolic Parametrization of Pipe and Canal Surfaces, ISSAC-2000 2000 ACM Press 194 200 7 A. K. Lenstra 8 M. Peternell H. Pottmann Computing rational parametrizations of canal surfaces J. Symb. Comput. 23 1997 255 266 9 J. Schicho Rational parameterization of real algebraic surfaces ISSAC-98 1998 ACM Press p. 302–308 10 J. Schicho Rational parametrization of surfaces J. Symb. Comput. 26 1998 1 30 11 J. Sendra F. Winkler Symbolic parametrization of curves J. Symb. Comput. 12 1991 607 632 12 J. Sendra F. Winkler Parametrization of algebraic curves over optimal field extensions J. Symb. Comput. 23 1997 191 208 13 P.S. Wang Factoring multivariate polynomials over algebraic number fields Math. Comput. 32 1976 324 336
Year
DOI
Venue
2001
10.1006/jsco.2001.0453
J. Symb. Comput.
Keywords
DocType
Volume
canal surface
Journal
32
Issue
ISSN
Citations 
1
Journal of Symbolic Computation
8
PageRank 
References 
Authors
0.59
5
3
Name
Order
Citations
PageRank
Günter Landsmann1121.76
Josef Schicho2897.31
Franz Winkler320726.00