Page 155 - Geometric Modeling and Algebraic Geometry

156 F. Cazals et al.

Fig. 8.5. Plot of the bi-quartic B` ezier surface with ridges

Φ (u, v)=[2/3 v +2/3 uv − 1/3 u v +1/3 v − 2/3 v u +1/3 u v ,
2 2
2
2
2
1/2 u +1/2 u + uv − u v − 1/2 v u +1/2 u v ,
2 2
2
2
2
1+3 v − u − 4 v +5 uv + u v − 7/2 v u − 5/2 u v ]
2
2 2
2
2
The ridge curve has total degree 56 and partial degrees 33 with 1078 terms and
coefficients with up to 15 digits. The computation of the biggest system of study
points S c takes 4.5 minutes. On this example, study point boxes have to be refined
up to a size of less than 2 −255 to compute the topology. The following table gives
the main characteristics of the study point systems:
System # of roots ∈ C # of real roots ∈D
70 1
S u
293 6
S p
695 5
S c