Proving and Solving over the Reals [SFB F1303-1]

@phdthesis{RISC2472,
author = {Ibolya Szilagyi},
title = {{Symbolic-Numeric Techniques for Cubic Surfaces}},
language = {english},
abstract = {In geometric modelling and related areas algebraic curves/surfacestypically are described either as the zero set of an algebraicequation (implicit representation), or as the image of amap given by rational functions (parametricrepresentation). The availability of both representations oftenresult in more efficient computations.Computational theories and techniques of algebraic geometry infloating point environment are of high interest in geometricmodelling related communities. Therefore, deriving approximatealgorithms that can be applied to numeric data have become a veryactive research area. In this thesis we focus on the twoconversion problems, called implicitization and parametrization,from the numeric point of view.A very important issue in the implicitization problem is theperturbation behavior of parametric objects. For a numericallygiven parametrization we cannot compute an exact implicitequation, just an approximate one. We introduce a conditionnumber of the implicitization problem to measure the worst effecton the solution, when the input data is perturbed by a smallamount. Using this condition number we study the algebraic andgeometric robustness of the implicitization process.Several techniques for parameterizing a rational algebraic surface as a whole exist. However, in many applications, itsuffices to parameterize a small portion of the surface. Thismotivates the analysis of local parametrizations, i.e.parametrizations of a small neighborhood of a given point $P$ ofthe surface $S$. We introduce several techniques for generatingsuch parameterizations for nonsingular cubic surfaces. For thisclass of surfaces, it is shown that the local parametrizationproblem can be solved for all points, and any such surface can becovered 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}
}

@phdthesis{RISC4133,
author = {Josef Schicho},
title = {{Rational Parametrization of Algebraic Surfaces. Symbolic Solution of an Equation in three Variables}},
language = {english},
year = {1995},
translation = {0},
school = {RISC, Johannes Kepler University Linz},
length = {0}
}