Page 98 - Geometric Modeling and Algebraic Geometry

96 T.-H. Lˆ e and A. Galligo
6.2.3 Parametric surfaces of bidegree (1, 2)

Definition 6. Parametric surfaces of bi-degree (1,2) are images of maps

Φ : P (C) × P (C) −→ P (C)
1
1
3
([t : s], [u : v]) −→ [Φ 1 : Φ 2 : Φ 3 : Φ 4 ]
where Φ 1 ,Φ 2 ,Φ 3 ,Φ 4 are bihomogeneous polynomials in [t : s] and [u : v] of bide-
gree (1, 2).
The parametric surfaces of bidegree (1, 2) are rational ruled surfaces and have
implicit degree 4 if Φ 1 ,Φ 2 ,Φ 3 ,Φ 4 have no base points. These (1, 2) parametric sur-
faces S are images of the normal surfaces F having minimum directrix conics and
generated by (1, 1) correspondence between 2 directrix conics. Hence, S is gener-
ated by (1, 1) correspondence between two non degenerated conics, or between a
double line and a conic, or between two double lines (respectively when the center
of projection is in general position in regard to F, or it cuts a plane containing a
directrix conic of F or it cuts two planes containing two directrix conics of F.
The implicit equations for each case are given in [12], pp (62-69). These equa-
tions contain many parameters. We aim to consider normal forms for bidegree (1,2)
parameterizations with a minimum number of parameters in the complex and in the
real settings.
We denote by (X : Y : Z : T) projective coordinates in P and by (X : Y : Z :
3
T : P : Q) projective coordinates in P .
5

6.3 Dual scroll

We write the (1, 2) parametric surfaces S in the basis

{tu , 2tuv, tv ,su , 2suv, sv },
2
2
2
2
⎧
⎪ X = a 1 tu +2b 1 tuv + c 1 tv + d 1 su +2e 1 suv + f 1 sv 2
2
2
2
⎪
⎨ Y = a 2 tu +2b 2 tuv + c 2 tv + d 2 su +2e 2 suv + f 2 sv 2
2
2
2
(S): (6.1)
⎪ Z = a 3 tu +2b 3 tuv + c 3 tv + d 3 su +2e 3 suv + f 3 sv 2
2
2
2
⎪
⎩
T = a 4 tu +2b 4 tuv + c 4 tv + d 4 su +2e 4 suv + f 4 sv 2
2
2
2
where a i ,b i ,c i ,d i ,e i ,f i ∈ C.
Notation: A is the 4×6 matrix of the coefficients a i ,b i ,c i ,d i ,e i ,f i . We can assume
that rank(A)=4.
The considered surface S can be seen either as the total space of a family of
1
conics S = ∪ t C t with t ∈ P (C), or as the total space of a family of lines S = ∪ u L u
with u ∈ P (C).
1