Page 144 - Geometric Modeling and Algebraic Geometry

8 Ridges and Umbilics of Polynomial Parametric Surfaces 145
8.1.4 Notations

Ridges and umbilics.

At any non umbilical point of the surface, the maximal (minimal) principal curvature
is denoted k 1 (k 2 ), and its associated direction d 1 (d 2 ). Anything related to the max-
imal (minimal) curvature is qualified blue (red), for example we shall speak of the
blue curvature for k 1 or the red direction for d 2 . Since we shall make precise state-
ments about ridges, it should be recalled that, according to definition 1, umbilics are
not ridge points.

Differential calculus.
For a bivariate function f(u, v), the partial derivatives are denoted with indices, for
3
example f uuv = ∂ f . The gradient of f is denoted f 1 or df =(f u ,f v ). The
∂ 2 u∂v
quadratic form induced by the second derivatives is denoted f 2 (u, v)= f uu u +
2
2f uv uv + f vv v . The discriminant of this form is denoted δ(f 2 )= f 2 − f uu f vv .
2
uv
The cubic form induced by the third derivatives in denoted f 3 (u, v)= f uuu u +
3
3f uuv u v +3f uvv uv + f vvv v . The discriminant of this form is denoted δ(f 3 )=
2
3
2
4(f uuu f uvv − f 2 )(f uuv f vvv − f 2 ) − (f uuu f vvv − f uuv f uvv ) .
2
uuv uvv
Let f be a real bivariate polynomial and F the real algebraic curve defined by f.
A point (u, v) ∈ C is called
2
• a singular point of F if f(u, v)=0, f u (u, v)=0 and f v (u, v)=0;
• a critical point of F if f(u, v)=0, f u (u, v)=0 and f v (u, v) =0 (such a point
has an horizontal tangent, we call it critical because if one fixes the v coordinate,
then the restricted function is critical wrt the u coordinate, this notion will be
useful in section 8.3);
• a regular point of F if f(u, v)=0 and it is neither singular nor critical.
If the domain D of study is a subset of R , one calls fiber a cross section of this
2
domain at a given ordinate or abscissa.
Misc.
The inner product of two vectors x, y is denoted x, y .

8.2 Relevant equations for ridges and its singularities

This section briefly recalls the equations defining the ridge curve and its singularities,
k
see [3]. Let Φ be the parameterization of class C for k ≥ 4. Denote I and II the
matrices of the first and second fundamental form of the surface in the basis (Φ u ,Φ v )
of the tangent space. In order for normals and curvatures to be well defined, we
assume the surface is regular i.e. det(I) =0.