Page 99 - Geometric Modeling and Algebraic Geometry

6 Classification of Surfaces 97
6.3.1 A 3-projective plane

Definition 7. We consider the map between dual spaces:
π A :(P ) → (P )
3 ∗
5 ∗
(α, β, γ, δ) → (A, C, E, B, D, F)=(α, β, γ, δ)A
t
defined by A. Its image is a 3-projective plane in (P ) that we denote by Π A .
5 ∗
5 ∗
By linear transformation, we can write the implicit equations of Π A in (P ) as
follows:

A 1 X 1 + B 1 X 2 + C 1 X 3 + D 1 X 4 + E 1 X 5 + F 1 X 6 =0
B 2 X 2 + C 2 X 3 + D 2 X 4 + E 2 X 5 + F 2 X 6 =0

where (X 1 : X 2 : X 3 : X 4 : X 5 : X 6 ) are projective coordinates of (P ) .
5 ∗
6.3.2 Tangent planes to all conics of the surface

3
We want to characterize the planes Π in P such that Π is tangent to any curve C (t:s)
of S or contains it.
The general equation of a plane Π in P is:
3
αX + βY + γZ + δT =0 (α, β, γ, δ) ∈ C \{0} (6.2)
4
Substituting in (6.2) the expressions of X, Y, Z, T given in (6.1), we obtain the equa-
tion of the intersection of Π and of a conic C (t:s) :

Π ∩C (t:s) :(At + Bs)u +2(Ct + Ds)uv +(Et + Fs)v =0.
2
2
where (A, C, E, B, D, F)= π A (α, β, γ, δ) ∈ (P ) .
5 ∗
They are tangent (or Π contains C (t:s) ) for all (t : s) ∈ P if and only if the
1
discriminant vanishes identically, i.e. (Ct + Ds) − (At + Bs)(Et + Fs)=0, ∀(t :
2
s) ∈ P . This is true if and only if the following conditions are satisfied:
1
C = AE , 2CD = AF + BE , D = BF.
2
2
From this, we obtain four simpler equations:
C = AE , D = BF , CD = AF = BE , (6.3)
2
2
(We note that the four equations above are related). We have:

AC B D
(6.3) ⇔ rank ≤ 1. (6.4)
CE D F
(6.4) defines a surface of (P ) , we denote by F(2, 2) (which is a so-called rational
5 ∗
∗
scroll. So we have the following proposition: