@**phdthesis**{RISC2472,author = {Ibolya Szilagyi},

title = {{Symbolic-Numeric Techniques for Cubic Surfaces}},

language = {english},

abstract = {In geometric modelling and related areas algebraic curves/surfaces
typically are described either as the zero set of an algebraic
equation (implicit representation), or as the image of a
map given by rational functions (parametric
representation). The availability of both representations often
result in more efficient computations.
Computational theories and techniques of algebraic geometry in
floating point environment are of high interest in geometric
modelling related communities. Therefore, deriving approximate
algorithms that can be applied to numeric data have become a very
active research area. In this thesis we focus on the two
conversion problems, called implicitization and parametrization,
from the numeric point of view.
A very important issue in the implicitization problem is the
perturbation behavior of parametric objects. For a numerically
given parametrization we cannot compute an exact implicit
equation, just an approximate one. We introduce a condition
number of the implicitization problem to measure the worst effect
on the solution, when the input data is perturbed by a small
amount. Using this condition number we study the algebraic and
geometric robustness of the implicitization process.
Several techniques for parameterizing a rational algebraic
surface as a whole exist. However, in many applications, it
suffices to parameterize a small portion of the surface. This
motivates the analysis of local parametrizations, i.e.
parametrizations of a small neighborhood of a given point $P$ of
the surface $S$. We introduce several techniques for generating
such parameterizations for nonsingular cubic surfaces. For this
class of surfaces, it is shown that the local parametrization
problem can be solved for all points, and any such surface can be
covered completely.},

year = {2005},

month = {July},

note = {PhD Thesis},

translation = {0},

school = {RISC-Linz},

keywords = {implicitization, numerical stability, local parametrization, cubic surface},

sponsor = {RISC PhD scholarship program of the government of Upper Austria, and by the Spezialforschungsbereich (SFB) grant F1303, Austrian Science Foundation (FWF).},

length = {112}

}