Page 95 - Geometric Modeling and Algebraic Geometry

6

General Classification of (1,2) Parametric Surfaces
3
in P

Thi-Ha Lˆ e and Andr´ e Galligo

Laboratoire J-A. Dieudonn´ e
Universit´ e de Nice Sophia-Antipolis
Parc Valrose, 06108 Nice Cedex 2, France
{lethiha, galligo}@math.unice.fr

Summary. Patches of parametric real surfaces of low degrees are commonly used in Com-
puter Aided Geometric Design and Geometric Modeling. However the precise description of
the geometry of the whole real surface is generally difficult to master, and few complete clas-
sifications exist.
Here we study surfaces of bidegree (1,2). We present a classification and a geometric
study of parametric surfaces of bidegree (1,2) over the complex field and over the real field by
considering a dual scroll. We detect and describe (if it is not void) the trace of self-intersection
and singular locus in the system of coordinates attached to the control polygon of a patch (1,2)
in the box [0; 1] × [0; 1].

6.1 Introduction

We consider a polynomial mapping of bidegree (1,2):

Φ : P × P −→ P 3
1
1
given by a matrix A =(a ij ), i =1,..., 4, j =1,..., 6 of maximal rank 4 such
that:
t
t
Φ = (Φ 1 ,Φ 2 ,Φ 3 ,Φ 4 )and Φ = A. (tu ,tuv,tv ,su ,suv,sv )(1)
2
2
2
2
where((t : s), (u : v)) are a system of coordinates of P × P . The base field is
1
1
K = C or R. Then S = Im(Φ) ⊂ P is a parametric surface of bidegree (1,2) and
3
Φ is a parameterization of S.
Similarly, one defines surfaces of bidegree (m, n); patches of these surfaces are
often used in C.A.G.D and Solid Modeling especially for the bi-cubics m = n =3.
Our aim is to classify the applications Φ of bidegree (1,2) while the base field is
R or C up to a change of projective coordinates in the source space P × P and in
1
1
the target space P . In a previous article [13] we described the generic complex case
3
and the geometry of the corresponding surfaces. Then, the parameterization of Φ is
equivalent to a parameterization, we called “normal form” and denoted by NF(a, b):