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Proof. Let V be the set of points p ∈ C, which project onto p′ . From the previous definition, we directly deduce that the projection of the tangent cone at p ∈ C is contained in the tangent cone of the projection of p. Thus the tangent cone of C ′ at p′ = (α, β) contains the projection of the tangent cones of the points p ∈ V . Since p ′ is regular, its tangent cone is a line parallel to the y direction. Therefore, the tangent cones of the points p ∈ V are in the plane x − α = 0, parallel to the plane (y, z).

Since the number of asymptotic directions of C is finite, by a generic linear change of variables, we can avoid the cases where C has an asymptotic direction parallel to the (y, z) plane. Next, we compute the x-critical points of C by solving the system (1), using algorithm 7. This computation allows us to check that the system is zero-dimensional and that the x-coordinate of the real solutions are distinct. If this is not the case, we perform a generic change of coordinates. The cases for which we have to do a change of coordinates are those where a component of C is in a plane parallel to (y, z) or where a plane parallel to (y, z) is tangent to C in two distinct points.

Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling an A. McKenney, S. Ostrouchov, and D. Sorensen. LAPACK Users’ Guide. SIAM, Philadelphia, 1992. org/lapack/. 3. S. Basu, R. -F. Roy. Algorithms in Real ALgebraic Geometry. SpringerVerlag, Berlin, 2003. ISBN 3-540-00973-6. 4. L. Bus´e, M. Elkadi, and B. Mourrain. Using projection operators in computer aided geometric design. In Topics in Algebraic Geometry and Geometric Modeling,, pages 321–342. Contemporary Mathematics, 2003.