# PhD Defence: Fat Arcs and Fat Spheres for Approximating Algebraic Curves and Solving Polynomial Systems

**PhD Defence: Fat Arcs and Fat Spheres for Approximating Algebraic Curves and Solving Polynomial Systems**

Studying objects defined by algebraic equations has been an active
research area for a long time. The reason for the interest is the wide
variety of applications, which appear in mathematical modeling and
physics. Modeling algebraic objects is an essential ingredient of
free-form surface visualization and numerical simulations. Thus modeling
algorithms are frequently used in CAD-systems, manufacturing, robotics
etc. Several problems in applications are described by multivariate
polynomial systems with a low dimensional solution set. In the thesis we
present a method to generate bounding regions for one- or
zero-dimensional solution sets of multivariate polynomial systems.

The one-dimensional solution set of a multivariate polynomial system
forms an algebraic curve. These curves are defined as the intersection
curves of algebraic surfaces. Representing these algebraic curves is a
fundamental problem of some geometric algorithms. For instance such
algebraic curves appear as the boundary curves of surfaces created by
Boolean operations or the self-intersection curves of surfaces. Due to
the importance of these curves several algorithms have been introduced
to approximate them, especially for curves embedded in lower dimensional
spaces. We formulate in the thesis a new geometrical method, which
approximates one-dimensional algebraic sets. The algorithm generates a
set of quadratic regions, the so called “fat arcs” , which encloses the
algebraic curve within a user specified tolerance. We describe different
methods, how to generate these bounding regions, and we study their
behavior. Then we combine the fat arc generation with the standard
subdivision technique.

The computation of zero-dimensional solution sets of multivariate
polynomial systems has also several applications in algebra and
geometry. Therefore various methods exist to find or to isolate the
roots of polynomial systems. They use symbolic, numeric or combined
techniques in order to find the solutions. In the end of the thesis we
generalize the definition of fat arcs to the concept of fat spheres. We
introduce an iterative domain reduction method based on fat sphere
generation. This method generates sequences of bounding regions, which
converge with order three to the single roots of a multivariate
polynomial system.