Revised papers presented at an ESF
Exploratory Workshop
(held at Schloß Weinberg, Kefermarkt
(Austria), September 29  October 3, 2003)
Ordering information 
Abstracts

Distance Separation Measures
Between Parametric Curves and Surfaces Toward Intersection and
Collision Detection Applications This paper
investigates the use of separation measures for parametric curves
and surfaces toward the resolution of interference and
intersections between curves and surfaces as well as collision
detection. Two types of distance separation measures are
discussed. While the trivial distance function can be derived
quite efficiently, it is shown in this work that this trivial
distance function is not the optimal approach, in general. better
and more efficient scheme that projects the distance onto the
normal field of either manifold is demonstrated to be superior in
correctly detecting highly coupled nonintersecting arrangements
as such. SelfIntersection Problems and
Approximate Implicitization We discuss how approximate implicit representations of parametric curves and surfaces may be used in algorithms for finding selfintersections. We first recall how approximate implicitization can be formulated as a linear algebra problem, which may be solved by an SVD. We then sketch a selfintersection algorithm, and discuss two important problems we are faced with in implementing this algorithm: What algebraic degree to choose for the approximate implicit representation, and for surfaces how to find selfintersection curves, as opposed to just points. Third Order Invariants of
Surfaces The
classical invariant theory from the 19th century is used to
determine a complete system of 3rd order invariants on a surface
in threespace. The invariant ring has 18 generators and the
ideal of syzygies has 65 generators. The invariants are expressed
as polynomials in the components of the first fundamental form,
the second fundamental form and the covariant derivative of the
latter, or in the case of an implicitly defined surface 
M=f^1(0)  as polynomials in the partial derivatives of f up to
order three. Polynomial C^2 Spline Surfaces
Guided by Rational Multisided Patches An
algorithm is presented for approximating a rational multisided
Mpatch by a C^2 spline surface. The motivation is that the
multisided patch can be assumed to have good shape but is in
nonstandard representation or of too high a degree. The algorithm
generates a finite approximation of the Mpatch, by a sequence of
patches of bidegree (5,5) capped off by patches of bidegree
(11,11) surrounding the extraordinary point. Elementary Theory of Del Pezzo
Surfaces Del Pezzo surfaces are certain algebraic surfaces in projective nspace of degree n. They contain an interesting configuration of lines and have a rational parametrization. We give an overview of the classification with an emphasis on algorithmic constructions (e.g. of the parametrization), on explicit computations, and on real algebraic geometry. On the Shape Effect of a Control
Point: Experimenting with NURBS Surfaces In this paper we further the theoretical investigation on the shape effect of a single control point, measured in terms of the sign of Gaussian and mean curvatures of the underlying parametric surface. The so far obtained theoretical results are illustrated by experimenting with three typical NURBS surfaces, namely a cylinder, an ellipsoid and a torus. Universal Rational Parametrizations
and Spline Curves on Toric Surfaces Recently a constructive description of all rational parametrizations for toric surfaces was described in terms of the universal rational parametrizations (URP). We give an elementary introduction to this theory from the Geometric Modelling point of view: toric surfaces are defined via homogeneous coordinates; projections, singular cases, and noncanonical real structures are described; the URP theorem is explained. A theory of rational C^1 spline curves with certain interpolation properties on toric surfaces is developed. Applications for smooth blending of natural quadrics are sketched. The Geometry of the Tangent
Developable The tangent developable of a curve C in IP^3$ is a singular surface with a cuspidal edge along C and the flex tangents of C. It also contains a multiple curve, typically double, and we express the degree of this curve in terms of the invariants of C. In many cases we can calculate the intersections of C with the multiple curve, and pictures of these cases are provided. Singularities of Some Projective
Rational Surfaces We discuss
the singularities of some rational algebraic surfaces in complex
projective space. In particular, we give formulas for the degrees
of the various types of singular loci, in terms of invariants of
the surface. A Recursive Taylor Method for
Algebraic Curves and Surfaces This paper
examines recursive Taylor methods for multivariate polynomial
evaluation over an interval, in the context of algebraic curve
and surface plotting as a particular application representative
of similar problems in CAGD. The modified affine arithmetic
method (MAA), previously shown to be one of the best methods for
polynomial evaluation over an interval, is used as a benchmark;
experimental results show that a second order recursive Taylor
method (i) achieves the same or better graphical quality compared
to MAA when used for plotting, and (ii) needs fewer arithmetic
operations in many cases. Furthermore, this method is simple and
very easy to implement. Distance Properties of
epsilonPoints on Algebraic Curves This paper deals with some mathematical objects that the authors have named epsilonpoints, and that appear in the problem of parametrizing approximately algebraic curves. This type of points are used as based points of the linear systems of curves that appear in the parametrization algorithms, and they play an important role in the error analysis. In this paper, we focus on the general study of distance properties of epsilonpoints on algebraic plane curves, and we show that if P* is an epsilonpoint on a plane curve C of proper degree d, then there exists an exact point P on C such that its distance to P* is at most sqrt(epsilon}) if P* is simple, and O(sqrt(epsilon)^{1/2d}) if P* is of multiplicity r>1. Furthermore, we see how these results particularize to the univariate case giving bounds that fit properly with the classical results in numerical analysis. Computing the Topology of
ThreeDimensional Algebraic Curves In this paper, we present a new method for computing the topology of curves defined as the intersection of two implicit surfaces. The main ingredients are projection tools, based on resultant constructions and 0dimensional polynomial system solvers. We describe a lifting method for points on the projection of the curve on a plane, even in the case of multiple preimages on the 3D curve. Reducing the problem to the comparison of coordinates of socalled critical points, we propose an approach which combines control and efficiency. An emphasis in this work is put on the experimental validation of this new method. Examples treated with the tools of the library axel (http://wwwsop.inria.fr/galaad/logiciels/axel/: Algebraic SoftwareComponents for gEometric modeLing) are showing the potential of such techniques. Challenges in SurfaceSurface
Intersections Tangential and singular situations are still challenges in a system for surfacesurface intersections. This paper presents several real world examples of hard intersection problems, and proposes methods on how to deal with them. In particular, solutions which use the possibility of representing a parametric surface as an algebraic surface through the use of approximate implicitization, are in focus. This allows us to transform an intersection between two parametric surfaces to the problem of finding zeroes of a function of two parameters. Numerical and Algebraic Properties
of Bernstein Basis Resultant Matrices Algebraic properties of the power and Bernstein forms of the companion, Sylvester and Bezout resultant matrices are compared and it is shown that some properties of the power basis form of these matrices are not shared by their Bernstein basis equivalents because of the combinatorial factors in the Bernstein basis functions. Several condition numbers of a resultant matrix are considered and it is shown that the most refined measure is NPhard, and that a simpler, suboptimal measure is easily computed. The transformation of the companion and B\'ezout resultant matrices between the power and Bernstein bases is considered numerically and algebraically. In particular, it is shown that these transformations are illconditioned, even for polynomials of low degree, and that the matrices that occur in these basis transformation equations share some properties. Approximate Parametrisation of
Confidence Sets In various
geometrical applications, the analysis and the visualization of
the error of calculated or constructed results is required. This
error has very often character of a nontrivial multidimensional
probability distribution. Such distributions can be represented
in a geometrically interesting way by a system of so called
confidence sets. In our paper we present a method for an
approximate parametrisation of these sets. 

Johannes Kepler University, Institute of Analysis, Dept. of Applied Geometry, Altenberger Str.69, 4040 Linz, Austria