Page 106 - Geometric Modeling and Algebraic Geometry

104 T.-H. Lˆ e and A. Galligo
6.5.2 Their intersection is infinite

ϕ 1 (t, u) ∩ ϕ 2 (t, u) is infinite if and only if ϕ 1 (t, u),ϕ 2 (t, u) have a common factor,
denoted by g(t, u) and it is not constant. So we can write:

ϕ 1 (t, u)= g(t, u)ψ 1 (t, u)
ϕ 2 (t, u)= g(t, u)ψ 2 (t, u)
We distinguish the following cases:
a) g(t, u) is of bidegree (1,0).
b) g(t, u) is of bidegree (1,1):
g(t, u) can be reduced.
g(t, u) cannot be reduced.
c) g(t, u) is of bidegree (0,2):
g(t, u) has a double root.
g(t, u) has two different roots.
d) g(t, u) is of bidegree (0,1).
The system {ψ 1 (t, u),ψ 2 (t, u)} has two different roots.
The system {ψ 1 (t, u),ψ 2 (t, u)} has a double root.

6.5.3 Parametric equations of the surface
We consider some particular cases and give the parametric equations of the surface
for each case. The remaining cases can be treated similarly.

Remark 13. We remind that the 3-projective plane Π A is defined by the transpose
matrix of the matrix A of the parameterization of the surface. The equations of Π A
can be written:

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
Π A :
A 2 X 1 + B 2 X 2 + C 2 X 3 + D 2 X 4 + E 2 X 5 + F 2 X 6 =0
We set:
ϕ 1 =(A 1 ,B 1 ,...,F 1 ) ∈ C \{0}
6
ϕ 2 =(A 2 ,B 2 ,...,F 2 ) ∈ C \{0}.
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Therefore,

t
Π A = {X = (X 1 ,...,X 6 ) ∈ C \{0}| (ϕ 1 ,X)=(ϕ 2 ,X)=0}.
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We observe that the rows of A are images of the points (1 : 0 : ... :0),... (0 : ... :
t
t
t
1) by A so, they belong to Π A . Hence kerA =< ϕ 1 , ϕ 2 >.
If rank A =4, we can transform A to the echelon form: