Page 104 - Geometric Modeling and Algebraic Geometry

102 T.-H. Lˆ e and A. Galligo
The proof is simple but tedious.
Therefore, by choosing 4 tangent planes as (X =0), (Y =0), (Z =0), (T =
0) and by a similar demonstration as in the generic complex case, we obtain the
parametric complex representation of the surface:
⎧
⎪ X = tu 2
⎪
⎨ iθ iθ
Y =(t − e s)(u − e v) 2
⎪ Z =(t − e −iθ s)(u − e −iθ v) 2
⎪
⎩
T = sv 2
We write the surface equations in the affine chart s = v = T =1 and take we have
that:
⎧
⎨ x = tu 2
y =(t − cos θ − i sin θ)(u − cos θ − i sin θ )

2
z =(t − cos θ + i sin θ)(u − cos θ + i sin θ )
⎩

2

By dividing y and z by sin θ sin θ and denoting a = cotan θ, b = cotan θ,we
2
obtain the following system:
⎧
⎪ x = tu 2
⎪
⎪ y t u
⎨
=( − a − i)( − b − i) 2
sin θ sin θ sin θ sin θ
2
⎪ z t u
⎪
⎪ =( − a + i)( − b + i) 2
⎩
sin θ sin θ sin θ sin θ
2
t u
By changing the parameters t = − a, u = − b and by transformation

sin θ sin θ
1

of coordinates (x ,y ,z )= (x, y, z) we obtain the surface equations as

sin θ sin θ
2
follows:
⎧
⎨ x =(t + a)(u + b) 2

y =(t − i)(u − i) 2

⎩
z =(t + i)(u + i) 2

y + z y − z

Finally, we transform (x ,y ,z )=(x , , ). Therefore we proved the

2 −2i
following proposition:
Proposition 10. A normal form for the parametric equations of a surface of type II
is as follows:
⎧
⎨ x =(t + a)(u + b) 2
(S): y = tu − t − 2u with a = cotan θ, b = cotan θ
2
z =2tu + u − 1
⎩
2
A surface of type II has two real pinch points corresponding to (t 1 = −b, u 1 =
.
−b), (t 2 = ∞,u 2 = ∞) and has two real torsal lines L u 1
and L u 2