Page 117 - Geometric Modeling and Algebraic Geometry

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In this part we collect six chapters which describe algorithms for geometric com-
puting with curves and surfaces.
Beck and Schicho discuss the parameterization of planar rational curves over
optimal field extensions, by exploiting the Newton polygon. Their method generates
a parameterization in a field extension of degree one or two.
Ridges and umbilics of surfaces are among the objects studied in classical dif-
ferential geometry, and they are of some interest for characterizing and analyzing
the shape of a surface. In the case of polynomial parametric surfaces, these special
curves are studied in the chapter by Cazals, Faug` ere, Pouget and Rouillier. In par-
ticular, the authors describe an algorithm which generates a certified approximation
of the ridges. In order to illustrate the efficiency, the authors report on experiments
where the algorithm is applied to B´ ezier surface patches.
Chau, Oberneder, Galligo and J¨uttler report on several symbolic-numeric tech-
niques for analyzing and computing the intersections and self-intersections of biqua-
dratic tensor product B´ ezier surface patches. In particular, they explore how far one
can go by solely using techniques from symbolic computing, in order to avoid po-
tential robustness problems.
Cube decompositions by eigenvectors of quadratic multivariate splines are ana-
lyzed by Ivrissimtzis and Seidel. The results are related to subdivision algorithms,
such as the tensor extension of the Doo–Sabin subdivision scheme.
A subdivision method for analyzing the topology of implicitly defined curves in
two- and three-dimensional space are studied by Liang, Mourrain and Pavone. The
method produces a graph which is isotopic to the curve. The authors also report on
implementation aspects and on experiments with planar curves, such as ridge curves
or self intersection curves of parameterized surfaces, and on silhouette curves of
implicitly defined surfaces.
The final chapter of this volume, by Shalaby and J¨uttler, describes techniques
for the approximate implicitization of space curves and of surfaces of revolution.
Both problems can be reduced to the planar situation. Special attention is paid to the
problem of unwanted branches and singular points in the region of interest.