Page 103 - Geometric Modeling and Algebraic Geometry

6 Classification of Surfaces 101
Each singular point (X(t : s),Y (t : s),Z(t : s),T(t : s)) is the intersection of
two lines L (u 1 :v 1 ) and L (u 2 :v 2 ) which belong to the plane Π (t:s) containing the conic
C (t:s) and to the surface, i.e. Π (t:s) ∩S = C (t:s) ∪L (u 1 :v 1 ) ∪L (u 2 :v 2 ) , where (u 1 :
v 1 ) , (u 2 : v 2 ) are roots of the equation:
[(b−a)t+a(1−b)s]su +2b(a−1)tsuv+(b−b )t v +(b−a)btsv =0. (6.5)
2
2
2 2
2
In the sequel, in order to simplify the readability, we shall often use affine coordinates
t instead of (t : s), u instead of (u : v) and so on.
6.4.2 The generic real case

Generically, the intersection of ϕ 1 (t, u) and ϕ 2 (t, u) is 4 distinct points:

(t 1 ; u 1 ), (t 2 ; u 2 ), (t 3 ; u 3 ), (t 4 ; u 4 ),
moreover all the t i (and all the u i ) are two by two distinct. These four points corre-
spond to four special tangent planes.

We observe that as the equations ϕ 1 (t, u) and ϕ 2 (t, u) have degree 1 in t,if u is
real then t is also real; and if u 1 ,u 2 are complex conjugate then the same holds for
t 1 and t 2 . So we have 3 cases that we denote by type I, type II, type III: either 4 real
points, or 2 real points and 2 conjugate points or two couples of conjugate points.
For all types, as in the generic complex case, each singular point is intersection of
where u 1 ,u 2 are the roots of an equation of degree 2 whose
and L u 2
two lines L u 1
coefficients are real polynomials in t. So either u 1 ,u 2 are reals or conjugate complex.
are two conjugate lines. Their intersection is always real.
and L u 2
Therefore L u 1
Hence, the singularity of the complex surface is real and moreover is a twisted cubic.
However only segments of this real twisted cubic form the singular locus of the real
parametric surface.
The study of the first case (type I) is as in the generic complex case. We present
the two last cases.

a) Two real and two conjugate points: type II

Lemma 9. We assume that t 1 ,t 2 ,u 1 ,u 2 ∈ R et t 3 ,t 4 ,u 3 ,u 4 ∈ C and t 3 = ¯ t 4 and
u 3 =¯u 4 . Hence, it exists two real homographies: η 1 ,η 2 : P (R)→P (R) and two
1
1

values θ, θ ∈ [0,π] such that:
iθ
η 1 (t 1 )=0,η 1 (t 2 )= ∞,η 1 (t 3 )= e ,η 1 (t 4 )= e −iθ
η 2 (u 1 )=0,η 2 (u 2 )= ∞,η 2 (u 3 )= e iθ ,η 2 (u 4 )= e −iθ