Partial Degree Formulae for Rational Algebraic Surfaces

J. R. Sendra, S. Perez-Diaz


In this paper, we present formulae for the computation of the partial degrees w.r.t. each variable of the implicit equation of a rational surface given by means of a proper parametrization. Moreover, when the parametrization is not proper we give upper bounds. These formulae generalize the results in (Sendra J.R., Winkler F. 2001: Tracing Index of Rational Curve Parametrizations. Computer Aided Geometric Design, Vol. 18, no. 8, pp. 771-795) to the surface case, and they are based on the computation of the degree of the rational maps induced by the projections, onto the coordinate planes of the three dimensional space, of the input surface parametrization. In addition, using the results presented in (Perez-Diaz S., Schicho J., Sendra J.R. 2002: Properness and Inversion of Rational Parametrizations of Surfaces. Applicable Algebra in Engineering, Communication and Computing, Vol. 13, pp.29-51) and (Perez-Diaz, S., Sendra, J.R., 2004: Computation of the Degree of Rational Surface Parametrizations. Journal of Pure and Applied Algebra. Vol. 193/1-3. pp. 99-121), the formulae simply involve the computation of the degree of univariate polynomials directed determined from the parametrization by means of some univariate resultants and some polynomial gcds.