Page 35 - Geometric Modeling and Algebraic Geometry

32 F. Aries et al.
good accuracy, and the eighteen degree surfaces associated to the bicubic parameter-
ization, which raise problems of complexity. Unfortunately, one may meet singular,
or close to singular parameterizations, that make computations unreliable. Thus one
needs to know as much as possible about the geometry of the space of quadratic
parameterizations.
The study of quadratic parameterizations is eased by considering, instead of the
affine setting, the projective setting. This means considering the projective quadratic
parameterizations of surfaces, that is the quadratic rational maps from the real pro-
jective plane RP to the real projective space RP . These maps are those of the form:
3
2
Ω ⊂ RP −→ RP 3 (2.2)
2
(x 0 : x 1 : x 2 ) −→ (f 0 : f 1 : f 2 : f 3 ) .
where the f i are quadratic forms in x 0 , x 1 , x 2 and Ω is a non–empty Zariski open
subset of RP .
2
The main topic of the present paper is the Invariant Theory of projective quadratic
parameterizations under linear changes of coordinates of RP and RP . Precisely, we
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2
provide a collection of covariants with simple geometric interpretation.
Let us give a motivating problem: the discrimination between the different kinds
of quadratic parameterizations of quartic surfaces. Let us make this precise. Con-
sider a quadratic map as in (2.2). Its image in RP is not, in general, Zariski–closed.
3
Consider its Zariski closure, it is an algebraic surface of degree at most 4. Let U
be the set of those maps for which it is a quartic, i.e. it has degree exactly 4.Two
elements of U are considered equivalent if one is obtained from the other by a linear
reparameterization (linear change of coordinates in the domain RP ) and a projective
2
transformation of the ambient space (linear change of coordinates in the codomain
RP ). Then, as it is shown in [7] and [8], there are finitely many equivalence classes
3
in U. The problem is to discriminate between these equivalence classes. Algorith-
mic solutions to this problem have been given in [2] and [7]. Our paper proposes a
new solution. It consists simply in providing polynomial equations and inequalities
3
defining the equivalence classes . The equivalence classes are actually orbits under
the action of some group. Thus it is natural to look for the equations and inequali-
ties among the objects provided by Classical Invariant Theory: the covariants. Then,
the aforementioned problem of discrimination between orbits of parameterizations
is solved as an application, by picking in our toolbox of covariants the most adapted
ones.
The sequel of the paper is organized as follows: Section 2.2 recalls known facts
about the classification of quadratic parameterizations of surfaces; Section 2.3 pro-
vides preliminaries on Classical Invariant Theory; Section 2.4 presents some geo-
metrical features of Steiner surfaces, that will be helpful to present our collection of
covariants; these covariants are introduced in Section 2.5; the last section, Section

Here is an example where the methods of [2] and [7] are not directly applicable: suppose
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we are given a family of parameterizations, depending on a parameter t. Then, by mere
specialization of the general equations and inequalities defining the classes, we are able to
determine which values of t give a parameterization in a given equivalence class.