Page 34 - Geometric Modeling and Algebraic Geometry

2

Some Covariants Related to Steiner Surfaces

Franck Aries , Emmanuel Briand , and Claude Bruchou 1
1
2
INRA Biom´ etrie, Avignon (France)
1
Franck.Aries@avignon.inra.fr Claude.Bruchou@avignon.inra.fr
Universidad de Cantabria, Santander (Spain)
2
emmanuel.briand@gmail.com
Summary. A Steiner surface is the generic case of a quadratically parameterizable quartic
surface used in geometric modeling. This paper studies quadratic parameterizations of sur-
faces under the angle of Classical Invariant Theory. Precisely, it exhibits a collection of co-
variants associated to projective quadratic parameterizations of surfaces, under the actions of
linear reparameterization and linear transformations of the target space. Each of these covari-
ants comes with a simple geometric interpretation.
As an application, some of these covariants are used to produce explicit equations and
inequalities defining the orbits of projective quadratic parameterizations of quartic surfaces.

2.1 Introduction

This paper deals with quadratically parameterizable quartic surfaces of R , that is
3
surfaces of degree 4 admitting a parameterization of the form:

R 2 −→ R 3
(2.1)
F 1 (x 1 ,x 2 ) F 2 (x 1 ,x 2 ) F 3 (x 1 ,x 2 )
(x 1 ,x 2 ) −→ , ,
F 0 (x 1 ,x 2 ) F 0 (x 1 ,x 2 ) F 0 (x 1 ,x 2 )
where the F i are polynomial functions of degree at most 2. For generic F i ’s, the
parameterized surface obtained is called a Steiner surface, see section 2.2 for the
precise definition.
Our general motivation for the study of Steiner surfaces is the following. Two
of us (Franck Aries and Claude Bruchou) are interested in mathematical modeling
of vegetation canopies (see [9] for more details). The detailed description of the ar-
chitecture of vegetation canopies is critical for the modeling of many agricultural
processes: the photosynthesis, the propagation of diseases from one organ to another
or the radiative transfer. These processes involve a big amount of computations on
geometric objects associated to each plant organ. Each geometric object can be ap-
proximated by a set of plane triangles, or more complex patches like bicubic. As
underlined in several papers of geometric modeling ([2, 7, 15]), Steiner patches are
a possibly good compromise between triangles, which need to be very many for a