Page 111 - Geometric Modeling and Algebraic Geometry

6 Classification of Surfaces 109
in the coordinates (x 1 ,y 1 ,z 1 ). The local equation of the surface becomes x − z 1 y .
2
2
1 1
Hence, P i is a pinch point also called a Whitney umbrella singularity in the real
setting. In our situation, the “rod of the umbrella” is curved, moreover the half line
is not visible in the real parametric surface.

Fig. 6.1. Whitney umbrella

We have the following proposition:
Proposition 14. 1) A non degenerate cubic curve of R does not admit an oval.
3
2) As a consequence, if the double locus in the parameter patch has an oval then
this oval contain two critical points (i.e pre-images of two pinch points).
Proof. 1) It is easy to prove.
2) We recall that the parameterization map Φ restricted to [0, 1] is continuous.
2
If the double locus in the parameter space has an oval O (hence a compact set) its
image by Φ is compact and is included in the singular locus C .As C is a twisted
cubic, it does not contain an oval, so the image of the oval must be a segment of curve
delimited by two points P 1 and P 2 . The pre-image of P 1 (respectively P 2 ) consists
of only one point (a critical point of Φ) which belongs to O. Hence, P 1 and P 2 are
two of the pinch points of the surface.

1

Φ P 1
E 1
u
Φ P 2
E 2

0
t 1
in R 3