Page 114 - Geometric Modeling and Algebraic Geometry

112 T.-H. Lˆ e and A. Galligo
are coincident. It is the self-intersection segment (MP) of the surface in

[0.6632643380; 1] × [0.1; 0.9]

(it is the same in [0; 1] × [0; 1]). The others segments in the figure (2) are phan-
tom curves: they correspond to double points of the parameterization Φ(t 1 ,u 1 )=
Φ(t 2 ,u 2 ) with (t 1 ,u 1 ) ∈ [0; 1] but (t 2 ,u 2 ) /∈ [0; 1] .
2
2
6.7 Conclusion

In this paper, we completed the classification of parametric surfaces of bidegree (1,2)
over the complex field and over the real field. In a future work,we will also provide
some results for the inverse problem: given a candidate (e.g. a segment of a line
or of twisted cubic curve), we look for a patch (1,2) which includes this candidate
as a subset of its singular locus. For instance we will characterize the ruled surfaces
containing a twisted cubic curve and such that all generating lines cut twice the cubic
curve, which are indeed parametric surfaces of bidegree (1, 2).

Acknowledgements

We would like to thank Ragni Piene and the anonymous referees for their com-
ments and suggestions. We acknowledge the partial support of the European Projects
GAIA II (IST-2001-35512) and of the Network of Excellence Aim@Shape (IST NoE
506766).

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