Page 33 - Geometric Modeling and Algebraic Geometry

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The second part of this book contains chapters which describe results concerning
special algebraic surfaces. Most surfaces used in geometric modeling are algebraic
surfaces of low degree, and their geometric nature, in particular their singularities,
can be analyzed using tools from real algebraic geometry. Here we collect several
results in this direction, which are organized in five chapters.
Aries, Briand and Bruchou analyze some covariants related to Steiner surfaces,
which are the generic case of a quadratically parameterizable quartic surface, fre-
quently used in geometric modeling. More precisely, they exhibit a collection of
covariants associated to projective quadratic parameterizations of surfaces with re-
spect to the actions of linear reparameterizations and linear transformations of the
target space. Along with the covariants, the authors provide simple geometric inter-
pretations. The results are then used to generate explicit equations and inequalities
defining the orbits of projective quadratic parameterizations of quartic surfaces.
The next chapter, authored by Breske, Labs and van Straten, is devoted to real
line arrangements and surfaces with many real nodes. It is shown that Chmutov’s
construction for surfaces with many singularities can be modified so as to give sur-
faces with only real singularities. The results show that all known lower bounds for
the number of nodes can be attained with only real singularities. The paper con-
cludes with an application of the theory of real line arrangements which shows that
the arrangements used by the authors are asymptotically the best possible ones for
the purpose of constructing surfaces with many nodes. This proves a special case of
a conjecture of Chmutov.
Johansen, Løberg and Piene study properties of monoid hypersurfaces – irre-
ducible hypersurfaces of degree d with a singular point of multiplicity d − 1. Since
such surfaces admit a rational parameterization, they are of potential interest in com-
puter aided geometric design. The main results include a description of the possible
real forms of the singularities on a monoid surface other than the (d − 1)-uple point.
The results are applied to the classification of singularities on quartic monoid sur-
faces, complementing earlier work on the subject.
The chapter by Krasauskas and Zube discusses canal surfaces which are gener-
ated as the envelopes of quadratic families of spheres. These surfaces generalize the
class of Dupin cyclides, but they are more flexible as blending surfaces between nat-
ural quadrics. The authors provide a classification from the point of view of Laguerre
geometry and study rational parameterizations of minimal degree, B´ ezier represen-
tations, and implicit equations.
Finally Lˆ e and Galligo present the classification of surfaces of bidegree (1,2)
over the fields of complex and real numbers. In particular, the authors study patches
of such surfaces, and they show how to detect and describe the loci in the parameter
domain – a [0, 1] × [0, 1] box – that map to selfintersections and singular points on
the surface.