Page 96 - Geometric Modeling and Algebraic Geometry

94 T.-H. Lˆ e and A. Galligo
X = tu ,Y =(t − s)(u − v) ,Z =(t − as)(u − bv) ,T = sv 2
2
2
2
where a and b are two complex parameters different from 0 and 1. Moreover, if

(a, b) =(a ,b ) then NF(a, b) is not equivalent to NF(a ,b ). We say that (a, b) is

a couple of moduli for this classification.
In this article we study the real generic cases and the non generic cases. The
surfaces S defined by (1) are ruled surfaces which admit an implicit equation in P 3
of degree at most 4. These surfaces were studied extensively in the 19th century by
great mathematicians: Cremona [7], Cayley [3], [4], Segre [28]; one finds a synthesis
of theirs results and extensions in the books of Salmon [25] and of Edge [12]. From
1930, the main stream of algebraic geometry concentrated on the study of varieties up
to birational equivalence and with more conceptual (and less effective) tools. How-
ever, applications in C.A.G.D and Solid Modeling showed the necessity of revisiting
of the geometry of parametrized curves and surfaces of small degrees and bidegrees.
An article of Coffman and al. [6] is a model of this kind of work: it revisited and
completed the classification of parameterized surfaces of total degree 2 (started by
Steiner in 1850). The ruled surfaces of implicit degree 4 are more complicated and
have more diversity. In the 19th century the focus was not on the classification of
parameterizations but rather on the geometric property and the calculation of cer-
tain invariants as well as on the obtaining of lists of implicit equations which are
dependent of many parameters. A presentation of these classification results over the
complex field related to rational (1,2)-B´ ezier surfaces with the description of the be-
haviour in presence of base-points, but without any description of the singularities,
was provided by W.L.F. Degen [9]. A more complete classification over the real field,
describing also the possible singularities was provided by S. Zube in [30] and [31].
Here we briefly review all these results, then we provide a new presentation based on
the study of the dual scroll and the consideration of the tangent planes to all conics
of the surface. We provide normal forms of the parameterizations and relate them to
geometric data of the surface. We also consider the problem of defining classifying
spaces which express the proximity with respect to deformations of these objects.
Our article is organized as following:
In section 2, we recall some results of the 19th century, we follow the syntax
given by Edge [12] in 1931, then we distinguish different types of parametric surfaces
and we concentrate on the surfaces of bidegree (1,2). In section 3, we present our
method of classification and introduce a scroll surface in the dual space which will be
used to find the moduli. This variety is different from those used by the geometricians
of 19th century but similar to the ones considered in [23]. In section 4, we recall
the results obtained in [13] for generic complex case and extend them to the real
setting. In section 5, we classify the intersections of a scroll (1,2) of P anda3-
5
projective plane or equivalently to the intersections of two curves of bidegree (1,2)
in P × P . Then we apply these results to the classification of parametric surfaces
1
1
(1,2). In section 6, we provide simple formulae to describe the critical points in the
system of coordinates attached to the control polygon of a patch (1,2). We detect and
describe the trace of the pre-images of the self-intersection and singular locus in the
box [0, 1] × [0, 1].