Catálogo de publicaciones - libros

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. In sect. 1 we describe our motivation, which consists in the study of the errors of so called Passive Observation Systems (POS). In sect. 2 we give a result about the intersection of quadric surfaces of revolution, which is useful in the investigation of the POS. In sect. 3 we give a general method for an approximate parametrisation of the confidence sets via simultaneous Taylor expansion. This method, which can be applied in a wide range of geometrical situations, is demonstrated on a concrete example of the POS.

Tangential and singular situations are still challenges in a system for surface-surface 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.

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 0-dimensional 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 so-called 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 (Algebraic Software-Components for gEometric modeLing) are showing the potential of such techniques.

This paper deals with some mathematical objects that the authors have named -points (see [8]), 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 -points on algebraic plane curves, and we show that if ⋆ is an -point on a plane curve of proper degree , then there exists an exact point on such that its distance to ⋆ is at most if ⋆ is simple, and () if ⋆ is of multiplicity > 1. Furthermore, we see how these results particularize to the univariate case giving bounds that fit properly with the classical results in numerical analysis.

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. A 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 non-intersecting arrangements as such.

Finally, a few extensions that further ease the detection of intersection-free arrangements, for both planar arrangements and arrangements in R, are also discussed.

Del Pezzo surfaces are certain algebraic surfaces in projective -space of degree . 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.

The tangent developable of a curve ⊂ ℙ is a singular surface with a cuspidal edge along and the flex tangents of . It also contains a multiple curve, typically double, and we express the degree of this curve in terms of the invariants of . In many cases we can calculate the intersections of with the multiple curve, and pictures of these cases are provided.

Algebraic properties of the power and Bernstein forms of the companion, Sylvester and Bézout 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 NP-hard, and that a simpler, sub-optimal measure is easily computed. The transformation of the companion and Bézout resultant matrices between the power and Bernstein bases is considered numerically and algebraically. In particular, it is shown that these transformations are ill—conditioned, even for polynomials of low degree, and that the matrices that occur in these basis transformation equations share some properties.

An algorithm is presented for approximating a rational multi-sided M-patch by a spline surface. The motivation is that the multi-sided 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 M-patch, by a sequence of patches of bidegree (5,5) capped off by patches of bidegree (11,11) surrounding the extraordinary point.

The philosophy of the approach is (i) that intricate reparametrizations are permissible if they improve the surface parametrization since they can be precomputed and thereby do not reduce the time efficiency at runtime: and (ii) that high patch degree is acceptable if the shape is controlled by a guiding patch.

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. We also consider which order of Taylor method is best to use, and propose that second order Taylor expansion is generally best. Finally, we briefly examine theoretically the relation between the Taylor method and the MAA method.