Implicitization and Parametrization of Nonsingular Cubic
Surfaces
Thomas Berry
berry@usb.ve
and
Richard R. Patterson
Department of Mathematical Sciences
Indiana University - Purdue University Indianapolis
402 North Blackford Street
Indianpolis, IN 46202-3216
(317) 274-6933
Fax (317) 274-3460
rpatters@math.iupui.edu
Abstract
In this paper we unify the two subjects of implicitization and parametrization
of nonsingular cubic surfaces. Beginning with a cubic parametrization with six
basepoints, we first form a three by four Hilbert-Burch matrix, and then
a three by three matrix of linear forms whose determinant is the implicit
equation. Beginning with an implicit equation, we show how to construct a
three by three matrix of linear forms whose determinant is the implicit
equation, and from it construct the Hilbert-Burch matrix and a
parametrization. The intermediate three by three matrix is shown to contain
information about lines and cubic curves that lie on the surface, as well as to
aid in the construction of inversion formulas.
File translated from TEX by TTH, version 2.52.
On 30 May 2000, 12:56.