Local parametrization of cubic surfaces

Abstract

Algebraic surfaces-which are frequently used in geometric modeling-are represented either in implicit or parametric form. Several techniques for parameterizing a rational algebraic surface as a whole exist. However, in many applications, it suffices to parameterize a small portion of the surface. This motivates the analysis of local parametrizations, i.e., parametrizations of a small neighborhood of a given point P of the surface S. In this paper we introduce several techniques for generating such parameterizations for nonsingular cubic surfaces. For this class of surfaces, it is shown that the local parametrization problem can be solved for all points, and any such surface can be covered completely.

On the number of Mordell–Weil generators for cubic surfaces

“…for details see [17,24]. The cubic surface S contains 3, 7, 15 or 27 real lines, and S(R) has either one or two connected components.…”

“…Let S be a smooth cubic surface defined over R. We quickly summarise some well-known facts; for details see [18] and [25]. The cubic surface S contains 3, 7, 15 or 27 real lines, and S(R) has either one or two connected components.…”