Page 154 - Geometric Modeling and Algebraic Geometry

8 Ridges and Umbilics of Polynomial Parametric Surfaces 155
methods and the computation time for the biggest system S c does not exceed 10
minutes with a Pentium M 1.6 Ghz. The following table gives the main characteris-
tics of these systems:

System # of roots ∈ C # of roots ∈ R # of real roots ∈D
160 16 8
S u
749 47 17
S p
S c 1432 44 19
In order to have more insight of the geometric meaning of the ridge curve, the
surface and its ridges are displayed on Fig. 8.5. This plot is computed without topo-
logical certification with the rs tci points function (from RS software, see also
[2]) from the polynomial P and then lifted on the surface.

Fig. 8.4. Bi-quartic B` ezier example : isotopic approximation of the ridge curve with study
points circled.

The second surface is a bi-quadratic B´ ezier surface