Page 141 - Geometric Modeling and Algebraic Geometry

142 F. Cazals et al.
that interestingly, (selected) ridges are also central in the analysis of Delaunay based
surface meshing algorithms [1].)
A comprehensive literature on ridges exists – see [11, 17, 18], and we just intro-
duce the basic notions so as to discuss our contributions. Consider a smooth embed-
ded surface whose principal curvatures are denoted k 1 and k 2 with k 1 ≥ k 2 .Away
from umbilical points — where k 1 = k 2 , principal directions of curvature are well
defined, and we denote them d 1 and d 2 . In local coordinates, we denote , the inner
product induced by the ambient Euclidean space, and the gradients of the principal
curvatures are denoted dk 1 and dk 2 . Ridges can be defined as follows — see Fig. 8.1
for an illustration :
Definition 1. A non umbilical point is called
• a blue ridge point if the extremality coefficient b 0 = dk 1 ,d 1 vanishes, i.e.
b 0 =0.
• a red ridge point if the extremality coefficient b 3 = dk 2 ,d 2 vanishes, i.e. b 3 =
0.
As the principal curvatures are not differentiable at umbilics, note that the extremality
coefficients are not defined at such points. Notice also the sign of the extremality
coefficients is not defined, as each principal direction can be oriented by two opposite
unit vectors. Apart from umbilics, special points on ridges are purple points – they
actually correspond to intersections between red and a blue ridges. The calculation
of ridges poses difficulties of three kinds.

Topological difficulties.

Ridges of a smooth surface feature self-intersections at umbilics — more precisely at
so-called 3-ridges umbilics — and purple points. From a topological viewpoint, re-
porting a certified approximation of ridges therefore requires reporting these singular
points.

Numerical difficulties.

As ridges are characterized by derivatives of principal curvatures, reporting them
requires evaluating third order differential quantities. Estimating such derivatives
depends upon the particular type of surface processed — implicitly defined, para-
meterized, discretized by a mesh, but is numerically a demanding task.

Orientation difficulties.

As observed above, the signs of the b 0 and b 3 depend upon the particular orientations
of the principal directions picked. But as a global coherent non vanishing orientation
of the principal directions cannot be found in the neighborhoods of umbilics, track-
ing the zero crossings of b 0 and b 3 faces a major difficulty. For the particular case of
surfaces represented by meshes, the so-called Acute rule can be used [4], but com-
puting meshes compliant with the requirements imposed by the acute rule is an open