Page 145 - Geometric Modeling and Algebraic Geometry

146 F. Cazals et al.
The principal directions d i and principal curvatures k 1 ≥ k 2 are the eigenvectors
and eigenvalues of the matrix W = I −1 II. The following equation defines coeffi-
cients A, B, C and D as polynomials wrt the derivative of the parameterization Φ up
to the second order
AB
= W(det I) 3/2 . (8.1)
CD
As a general rule, in the following calculations, we will be interested in deriving
quantities which are polynomials wrt the derivatives of the parameterization. These
calculations are based on quantities (principal curvatures and directions) which are
independent of a given parameterization, hence the derived formula are valid for any
parameterization.
Umbilics are characterized by the equation p 2 =0, with p 2 =(k 1 − k 2 ) (det I) .
2
3
We then define two vector fields v 1 and w 1 orienting the principal direction field d 1
√
v 1 =(−2B, A − D − p 2 )
√
w 1 =(A − D + p 2 , 2C).

Derivatives of the principal direction k 1 wrt these two vector fields define a, a ,b,b
by the equations:
√ √ √ √
a p 2 + b = p 2 (det I) 5/2 dk 1 ,v 1 ; a p 2 + b = p 2 (det I) 5/2 dk 1 ,w 1 .

(8.2)
The following definition is a technical tool to state the next theorem in a simple
way. The function Sign ridge introduced here will be used to classify ridge colors.
Essentially, this function describes all the possible sign configurations for ab and a b

at a ridge point.
Definition 2. The function Sign ridge takes the values

ab < 0 ab ≤ 0
-1 if or ,
a b ≤ 0 a b < 0

ab > 0 ab ≥ 0
+1 if or ,
a b ≥ 0 a b > 0

0 if ab = a b =0.

Theorem 3. The set of blue ridges union the set of red ridges union the set of umbilics
has equation P =0 where P =(a p 2 −b )/B is a polynomial wrt A, B, C, D, det I
2
2
as well as their first derivatives and hence is a polynomial wrt the derivatives of the
parameterization up to the third order. For a point of this set P, one has:
• If p 2 =0, the point is an umbilic.
• If p 2 =0 then:
–if Sign ridge = −1 then the point is a blue ridge point,
–if Sign ridge =+1 then the point is a red ridge point,
–if Sign ridge =0 then the point is a purple point.