@inproceedings{10.1007/3-540-54195-0_55,
    Abstract = {In this paper we use Gr{\"o}bner bases for the implicitization of rational parametric curves and surfaces in 3D-space. We prove that the implicit form of a curve or surface given by the rational parametrization {\$}{\$}x{\\_}1 : = {\backslash}frac{\{}{\{}p{\\_}1 {\}}{\}}{\{}{\{}q{\\_}1 {\}}{\}}x{\\_}2 : = {\backslash}frac{\{}{\{}p{\\_}1 {\}}{\}}{\{}{\{}q{\\_}2 {\}}{\}}x{\\_}3 : = {\backslash}frac{\{}{\{}p{\\_}1 {\}}{\}}{\{}{\{}q{\\_}3 {\}}{\}}{\$}{\$}where the p's and q's are univariate polynomials in y1or bivariate polynomials in y1, y2over a field K, can always be found by computing {\$}{\$}GB({\backslash}{\{} q{\\_}{\{}^{\{}{\\_}1 {\}} {\}} {\backslash}cdot x{\\_}1 - p{\\_}{\{}1,{\}} q{\\_}2 {\backslash}cdot x{\\_}2 - p{\\_}{\{}2,{\}} q{\\_}3 {\backslash}cdot x{\\_}3 - p{\\_}{\{}3,{\}} {\backslash}{\}} ) {\backslash}cap K[x{\\_}1 ,x{\\_}2 ,x{\\_}3 ],{\$}{\$}where GB is the Gr{\"o}bner basis with respect to the lexical ordering with x1≺x2≺x3≺y1≺y2, if for every i, j∈{\{}1,2,3{\}} with i≠j the polynomials pi, qi,pj,qjhave no common zeros. This result leads immediately to an implicitization algorithm for arbitrary rational parametric curves.},
    Address = {Berlin, Heidelberg},
    Author = {Kalkbrener, Michael},
    BookTitle = {Applied Algebra, Algebraic Algorithms and Error-Correcting Codes},
    Editor = {Sakata, Shojiro},
    File = {Implicitization of Rational Parametric Curves and Surfaces - Kalkbrener1991\_Chapter\_ImplicitizationOfRationalParam - a - l.pdf},
    ISBN = {978-3-540-47489-0},
    Pages = {249--259},
    Publisher = {Springer Berlin Heidelberg},
    Title = {Implicitization of rational parametric curves and surfaces},
    Year = {1991},
    date-added = {2020-11-02 16:16:42 +0100},
    date-modified = {2020-11-02 16:16:42 +0100},
    file-2 = {Implicitization of Rational Parametric Curves and Surfaces - publication\_1077 - a - l.pdf},
    file-3 = {Implicitization of Rational Parametric Curves and Surfaces - Kalkbrener90 - a - l.pdf},
    doi = {10.1007/3-540-54195-0_55}
}