Page 151 - Geometric Modeling and Algebraic Geometry

152 F. Cazals et al.
v

Number of
branches
above
v 2 i
+
n i,j
α i
Number of
branches
v 1 i
below
−
n i,j
1 2 1 2 2 u
β i,1 β i,1 u i,1 u i,1 1 2 1
β i,l i β i,l i u i,m i u i,m i
Fig. 8.2. Notations for a fiber involving several critical/singular points: u 1(2) are used for study
i,j
points, β i,j for simple points.
1(2)

v

α i+1

δ i

v 2 i
α i
v 1 i

u

Fig. 8.3. Performing connections between the study point fiber α i and the intermediate fiber
δ i

8.3.6 Step 2. Regularization of the study boxes

At this stage, we have computed isolating boxes of all study points {q i,j ,i =
1 ...s,j =1 ...m i } :the v-coordinates α 1 ,...,α s are isolated by intervals [v ; v ],
2
1
i i
i =1 ...s and the u-coordinates of the m i study points in each fiber α i are isolated
by intervals [u ; u ],j =1 ...m i .
1
2
i,j i,j
We know the number of branches of the curve passing through each study point :
it is 6 for a 3-ridge umbilic, 4 for a purple and 2 for others. We want to compute the
number branches coming from the bottom and from the top. We first reduce the box
until the number of intersections between the curve and the border of the box matches