Page 110 - Geometric Modeling and Algebraic Geometry

108 T.-H. Lˆ e and A. Galligo
M z
=((e + a − 1 − d − b)u +(2b +1 − 2e)u − b + e)t+(4bd − 1+2e +2a
2
C(t, u)
− 4ea)u +(d − 5e +2 − 2a +6ea − 6bd)u +(2bd +4e − 2ea − 1)u−e.
3
2
M x M y M z
By computation we check that the vector [ , , ] is equal to the
C(t, u) C(t, u) C(t, u)
∂Φ(t, u) ∂Φ(t, u)
cross-product ∧ .
∂t ∂u
∂Φ(t, u) ∂Φ(t, u)
We consider the points (t, u) for which ∧ vanishes, i.e the
∂t ∂u
M x M y M z
common roots of , , . More precisely, (t, u) is a root of the
C(t, u) C(t, u) C(t, u)
system:
0=(2af − e +1+2ea − 2bf − f + b − c − 2bd − 2a +2ec − 2cd)u 3 (6.7)
+(f − 2 − 5ec +5bd +2c − 5ea +5bf − af − 3b +3a +3e + cd)u 2
+(1 − 3e − 4bf +4ec − a − c − 4bd +4ea +3b)u+e+bd+bf−ec−ea−b
((2a − 4af +4cd +2f − 1)u −(2cd +3f − 2af +2a + d − 2)u+d+f−1
2
t =
(a + c − d − f)(u − 1)
Generically, the system (6.7) has 4 roots (a root is (∞, ∞). They are the critical
points of the parameterization and belong to the closure of the double locus C(t, u)
in the parameter space. We denote them by E 1 ,E 2 ,E 3 ,E 4 .

1 containing the conic C t 0 of the
For each t 0 ∈ P , we calculate the plane Π t 0
∩S is determined by an equation (t − t 0 )g(t 0 ,u)=0,
surface. The intersection Π t 0
where g(t 0 ,u) is polynomial of bidegree (2,2) in (t 0 ,u). If we consider it as poly-
∩S =
nomial of degree 2 in u so g(t 0 ,u) has two roots u 1 ,u 2 . Hence Π t 0
. The polynomial g(t 0 ,u) has a double root in u if its discrimi-
∪L u 1
∪L u 2
C t 0
nant with respect to u vanishes and generically it vanishes for 4 values of t 0 .For
each one of these values we have a corresponding value of u depending on t 0 and
being a double root of g(t 0 ,u), hence we obtain 4 corresponding values of u (c.f
[13]). Therefore the surface have 4 torsal lines corresponding to these four values
of u. By replacing each of these values in the equation C(t, u) we obtain 4 critical
points of the parameterization. They are actually the 4 points E 1 ,E 2 ,E 3 ,E 4 .We
denote their images by Φ respectively by P 1 ,P 2 ,P 3 ,P 4 .
Hence, in the general case, the singular locus of S consists in a twisted cubic C
(the closure of the image of the double locus C(t, u) of the parameterization) and
4 embedded points P 1 ,P 2 ,P 3 ,P 4 in C . Near P 1 (or P 2 ,P 3 ,P 4 ) the surface is de-
intersecting in a point P of
and L u 2
scribed by a continuous family of two lines L u 1
C . But when P ≡ P 1 the lines coincide and we obtain a torsal line of the surface.
By the implicit function theorem, we can apply a local isomorphism of R at
3
P i (i =1,..., 4) which transform locally the curve C into a line (x 1 =0,y 1 =0)