Catálogo de publicaciones - libros

We discuss how approximate implicit representations of parametric curves and surfaces may be used in algorithms for finding self-intersections. 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 self-intersection 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 self-intersection , as opposed to just points.

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. These enumerative results can be used, on the one hand, to show the existence of singularities in the complex case, and, on the other hand, as an “upper bound” for the singularities that can occur on a rational surface.

In this paper we further the theoretical investigation, initiated in [1], 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 surfaces, namely a cylinder, an ellipsoid and a torus.

The classical invariant theory from the 19th century is used to determine a complete system of 3rd order invariants on a surface in three-space. 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 — = (0) — as polynomials in the partial derivatives of up to order three.

As an application some commonly used fairings measures are written in invariant form. It is shown that the ridges and the subparabolic curve of a surface are the zero set of invariant functions and it is finally shown that the Darboux classification of umbilical points can be given in terms of two invariants.

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 non-canonical real structures are described; the URP theorem is explained. A theory of rational spline curves with certain interpolation properties on toric surfaces is developed. Applications for smooth blending of natural quadrics are sketched.

The synopsis refers to the interactions between The synopsis is based on a numerical example. The initial output gap in Germany is 60, and the initial output gap in France is 30. As a result, taking the sum over all periods, Table 8.1 shows: Obviously, the result depends on the type of policy system.