Page 19 - Geometric Modeling and Algebraic Geometry

1 The GAIA Project 13
• Development of prototypes of tools for constructing, manipulating and exploiting
implicit representations for parametric curves and surfaces based on resultant
computations.
One paper from the project addressing resultants is [7].

1.5.2 Singularities

Understanding the singularities of algebraic curves and surfaces is important for un-
derstanding the geometry of these curves and surfaces. A difficult problem in CAGD
is the handling of self-intersections, and the theory of singularities of algebraic va-
rieties is potentially a tool for handling this problem. In the GAIA project special
emphasis has been put on detecting and locating singularities appearing on parame-
terized and implicitly given curves and surfaces of low degree.

• The presence of singularities affects the geometry of complex and real projective
hypersurfaces and of their complements. We have illustrated the general princi-
ples and the main results by many explicit examples involving curves and sur-
faces.
• We have classified and analyzed the singularities of a surface patch given by a
parameterization in order to proceed to an early detection. We distinguish alge-
braically defined surface patches and procedural surfaces given by evaluation of
a program. Also we distinguish between singularities which can be detected by a
local analysis of the parameterization and those which require a global analysis,
and are more difficult to achieve.
• The detection of singularities is a critical ingredient of many geometrical prob-
lems, in particular in intersection operations. Once these critical points are
located, one can for instance safely use numerical methods to follow curve
branches. Detecting a singularity in a domain may also help in combining several
types of methods.

A paper addressing singularities from the project is [48].

1.5.3 Classification

To use algebraic curves and surfaces in CAGD one needs to know about their
shape: topology, singularities, self-intersections, etc. Most of this kind of classifica-
tion theory is performed for algebraic curves and surfaces defined over the complex
numbers, i.e., one considers complex (instead of only real) solutions to polynomial
equations in two or three variables (or in three or four homogeneous variables, if
the curves and surfaces are considered in projective space). Complete classification
results exist only for low degree varieties (implicit curves and surfaces) and mostly
only in the complex case. A simple example, the classification of conic sections, il-
lustrates well that the classification over the real numbers is much more complicated
than over the complex numbers.