Computing with curves and surfaces is ubiquitous in Computer Aided
Design and Modelisation. Most of the modeller softwares nowadays use
parametric representations for computing with such geometric objects.
We are however witnessing the emergence of a new trend of methods, which are
based on implicit representations of curves and surfaces.
Indeed, patches of implicit surfaces are effectively used to modelise
complex scenes, providing more compact or synthetic data structures
than those based on parametric representation. This is particularly important
for instance when one need to construct huge 3D-model models or when these
models have to be exchanged through the net.
However, these new representations also rise new problems, concerning the
manipulation of such objects.
Computer algebra (and more precisely effective algebraic geometry) is
offering a framework for handling such problems. The recent developments
of efficient algorithms for solving polynomial equations, allow us to
consider from a new point of view, questions which were previously
eluded or not treated. The objective of this work is to evaluate the impact
of such methods on geometric modelisation. It will consists in
developing a solver devoted to the manipulations of implicit surfaces (with
a special concern for quadratic surfaces). We will address in particular
the problems of
- computing efficiently and robustly the intersection points of three
surfaces.
- computing the intersection curve of two surfaces.
- specifying and implementing certificated predicates on these points of
intersection.