Page 105 - Geometric Modeling and Algebraic Geometry

6 Classification of Surfaces 103
b) Two conjugate couples: type III

Lemma 11. We assume that (t 1 ,u 1 )= (t 2 ,u 2 ) and (t 3 ,u 3 )= (t 4 ,u 4 ). It exists two

real homographies η 1 ,η 2 : P (R)→P (R) and two values θ, θ ∈ [0,π] such that:
1
1
iθ
η 1 (t 1 )= i, η 1 (t 2 )= −i, η 1 (t 3 )= e ,η 1 (t 4 )= e −iθ
η 2 (u 1 )= i, η 2 (u 2 )= −i, η 2 (u 3 )= e iθ ,η 2 (u 4 )= e −iθ
Therefore, by choosing the four special tangent planes as (X =0), (Y =
0), (Z =0), (T =0) and by a similar demonstration as in the generic complex
case, we have the parametric complex representation of the surface in the affine chart
s = v =1:
⎧
⎪ X =(t − i)(u − i) 2
⎪
Y =(t + i)(u + i) 2
⎨
iθ
⎪ Z =(t − e )(u − e iθ ) 2
⎪
⎩ −iθ −iθ
T =(t − e )(u − e ) 2
By similar transformation as in the case (a), we obtain the following proposition
(with two moduli θ and θ ):

Proposition 12. A surface of type III has a real parameterization as follows:
⎧
X = tu − t − 2u
2
⎪
⎪
⎪ Y =2tu + u − 1
⎪
2
⎨
(S): Z =( t − cotan θ)(( u − cotan θ ) − 1) − 2( u − cotan θ )

2
⎪ sin θ sin θ sin θ
⎪
⎪ t u u
⎪
T =2( − cotan θ)( − cotan θ )+( − cotan θ ) − 1
⎩ 2
sin θ sin θ sin θ
6.5 Non generic cases
We now list the following particular cases arising in the intersection of two curves of
bidegree (1,2) whose equations are ϕ 1 (t, u) and ϕ 2 (t, u).
6.5.1 Their intersection is finite
Set ϕ 1 (t, u) ∩ ϕ 2 (t, u)= {(t 1 ,u 1 ), (t 2 ,u 2 ), (t 3 ,u 3 ), (t 4 ,u 4 )}. We distinguish the
following cases:
a) 4 distinct points.
We have two cases: either (t 1 = t 2 and t 3 = t 4 )or(t 1 = t 2 and t 3 = t 4 ).
b) 2 distinct points and 1 double point (t 3 ,u 3 )=(t 4 ,u 4 ).
We have 3 cases: either (t 1 − t 2 )(t 2 − t 3 )(t 1 − t 3 ) =0,or t 1 = t 2 or
t 1 = t 3 (= t 4 ).
c) 2 double points.
d) 1 triple point and 1 simple point.