Page 100 - Geometric Modeling and Algebraic Geometry

98 T.-H. Lˆ e and A. Galligo
Proposition 8. A plane Π defined by (α, β, γ, δ) in P is tangent to all conics of S
3
∗
(or contains) it if and only if π A (α, β, γ, δ) ∈ F(2, 2) .
∗
In the following section, we express F(2, 2) as the dual scroll (in a geometric
sense that we will make precise) of the scroll F(2, 2) and construct related parametric
∗
equations for F(2, 2) .

6.3.3 Parameterization of the dual scroll

Notations: We use affine coordinates t instead of (t : s), u instead of (u : v).Weset
the following notation and parametric equations of the scroll F(2, 2) in P (it is the
5
normal ruled surface of bidegree (1,2)):
⎧
⎪ X = tu 2
⎪
⎪
⎪ Y = tu
⎪
⎪
⎨
Z = t
F(2, 2) :
⎪ T = u 2
⎪
⎪
⎪ P = u
⎪
⎪
⎩
Q =1
F(2, 2) is a surface and not a hypersurface. However for each point of F(2, 2) we
want to construct a hyperplane naturally attached to that point (This process is some-
how similar to the construction of the osculating plane attached to a point of a space
curve). These hyperplanes will describe a projective variety that we call the “dual
scroll” in (P ) . This is not related to the usual but to a generalized notion of duality,
5 ∗
already studied in [23] and called “strict duality”.
Construction: We consider the affine chart Q =1 where F(2, 2) becomes an affine
complete intersection, then its affine implicit equations are:
⎧
⎨ X − TZ =0
Y − ZP =0
⎩
T − P 2 =0
We denote by M the parameterization map of the scroll F(2, 2). To each point
M 0 = M(t 0 ,u 0 ),((t 0 ,u 0 ) =(0, 0)) of the scroll, we associate generalized tangent
(that are the generator
and C t 0
spaces of dimension 3 and 4 constructed from L u 0
and the conic of the scroll passing through M 0 ).
are:
and of the conic C t 0
The parametric equations of the line L u 0
⎧ ⎧
⎪ X = tu 2 ⎪ X = t 0 u 2
⎪ 0 ⎪
⎪ ⎪
⎪ ⎨ Y = t 0 u
⎪
⎨ Y = tu 0
: Z = t C t 0 : Z = t 0
L u 0
⎪ T = u
⎪ T = u 2 ⎪ 2
⎪
⎪
⎪
⎪ 0 ⎪
⎩ ⎩
P = u
P = u 0