Page 142 - Geometric Modeling and Algebraic Geometry

8 Ridges and Umbilics of Polynomial Parametric Surfaces 143
problem. For surfaces represented implicitly or parametrically, one can resort to the
Gaussian extremality E g = b 0 b 3 , which eradicates the sign problems, but prevents
from reporting the red and blue ridges separately.

Fig. 8.1. Umbilics, ridges, and principal blue foliation on the ellipsoid for normals pointing
outward

8.1.2 Previous work
Given the previous difficulties, no algorithm reporting ridges in a certified fashion
had been developed until this work. Most contributions deal with sampled surfaces
known through a mesh, and a complete review of these contributions can be found
in [4]. In the following, we focus on contributions related to parametric surfaces.
Reporting umbilics. Umbilics of a surface are always traversed by ridges, so that
reporting ridges faithfully requires reporting umbilics. To do so, Morris [13] mini-
mizes the function k 1 − k 2 , which vanishes exactly at umbilics. Meakawa et al. [15]
define a polynomial system whose roots are the umbilics. This system is solved with
the rounded interval arithmetic projected polyhedron method. This algorithm uses
specific properties of the Bernstein basis of polynomials and interval arithmetic. The
domain is recursively subdivided and a set of boxes containing the umbilics is output,
but neither existence nor uniqueness of an umbilic in a box is guaranteed.
Reporting ridges. The only method dedicated to parametric surfaces we are aware of
is that of Morris [13, 14]. The parametric domain is triangulated and zero crossings
are sought on edges. Local orientation of the principal directions are needed but only
provided with a heuristic. This enables to detect crossings assuming (i) there is at
most one such crossing on an edge (ii) the orientation of the principal directions is
correct. As this simple algorithm fails near umbilics, these points are located first
and crossings are found on a circle around the umbilic.
Equation of the ridge curve. Ridges can be characterized either as extrema of prin-
cipal curvatures along their curvature lines as in definition 1, or by analyzing the con-
tact between the surface and spheres [11]. For parametric surfaces, this later approach