On the implicit equation of conics and quadrics offsets

A new determinantal representation for the implicit equation of offsets to conics and quadrics is derived. It is simple, free of extraneous components and provides a very compact expanded form, these representations being very useful when dealing with geometric queries about offsets such as point positioning or solving intersection purposes. It is based on several classical results in ?A Treatise on the Analytic Geometry of Three Dimensions? by G. Salmon for offsets to non-degenerate conics and central quadrics.This research was funded by the Spanish Ministerio de Economía y Competitividad and by the European Regional Development Fund (ERDF), under the project MTM2017-88796-P

Rational conchoid and offset constructions: algorithms and implementation

This paper is framed within the problem of analyzing the rationality of the components of two classical geometric constructions, namely the offset and the conchoid to an algebraic plane curve and, in the affirmative case, the actual computation of parametrizations. We recall some of the basic definitions and main properties on offsets (see [13]), and conchoids (see [15]) as well as the algorithms for parametrizing their rational components (see [1] and [16], respectively). Moreover, we implement the basic ideas creating two packages in the computer algebra system Maple to analyze the rationality of conchoids and offset curves, as well as the corresponding help pages. In addition, we present a brief atlas where the offset and conchoids of several algebraic plane curves are obtained, their rationality analyzed, and parametrizations are provided using the created packages

Rational Generalized Offsets of Rational Surfaces

The rational surfaces and their offsets are commonly used in modeling and manufacturing. The purpose of this paper is to present relationships between rational surfaces and orientation-preserving similarities of the Euclidean 3-space. A notion of a similarity surface offset is introduced and applied to different constructions of rational generalized offsets of a rational surface. It is shown that every rational surface possesses a rational generalized offset. Rational generalized focal surfaces are also studied

Design and Implementation of Conchoid and Offset Processing Maple Packages

Annals of Mathematics and Artificial Intelligence presents a range of topics of concern to scholars applying quantitative, combinatorial, logical, algebraic and algorithmic methods to diverse areas of Artificial Intelligence, from decision support, automated deduction, and reasoning, to knowledge-based systems, machine learning, computer vision, robotics and planning

Partial Degree Formulae for Plane Offset Curves

In this paper we present several formulae for computing the partial degrees of the defining polynomial of the offset curve to an irreducible affine plane curve given implicitly, and we see how these formulae particularize to the case of rational curves. In addition, we present a formula for computing the degree w.r.t the distance variable.Comment: 24 pages, no figure

Reparametrizing Swung Surfaces over the Reals

Let K⊆R be a computable subfield of the real numbers (for instance, Q). We present an algorithm to decide whether a given parametrization of a rational swung surface, with coefficients in K(i), can be reparametrized over a real (i.e., embedded in R) finite field extension of K. Swung surfaces include, in particular, surfaces of revolution

Properness and inversion of rational parametrizations of surfaces

In this paper we characterize the properness of rational parametrizations of hypersurfaces by means of the existence of intersection points of some additional algebraic hypersurfaces directly generated from the parametrization over a field of rational functions. More precisely, if V is a hypersurface over an algebraically closed field ? of characteristic zero and is a rational parametrization of V, then the characterization is given in terms of the intersection points of the hypersurfaces defined by x i q i (t¯)−p i (t¯), i=1,...,n over the algebraic closure of ?(V). In addition, for the case of surfaces we show how these results can be stated algorithmically. As a consequence we present an algorithmic criteria to decide whether a given rational parametrization is proper. Furthermore, if the parametrization is proper, the algorithm also computes the inverse of the parametrization. Moreover, for surfaces the auxiliary hypersurfaces turn to be plane curves over ?(V), and hence the algorithm is essentially based on resultants. We have implemented these ideas, and we have empirically compared our method with the method based on Gröbner basis

Using implicit equations of parametric curves and surfaces without computing them: Polynomial algebra by values

The availability of the implicit equation of a plane curve or of a 3D surface can be very useful in order to solve many geometric problems involving the considered curve or surface: for example, when dealing with the point position problem or answering intersection questions. On the other hand, it is well known that in most cases, even for moderate degrees, the implicit equation is either difficult to compute or, if computed, the high degree and the big size of the coefficients makes extremely difficult its use in practice. We will show that, for several problems involving plane curves, 3D surfaces and some of their constructions (for example, offsets), it is possible to use the implicit equation (or, more precisely, its properties) without needing to explicitly determine it. We replace the computation of the implicit equation with the evaluation of the considered parameterizations in a set of points. We then translate the geometric problem in hand, into one or several generalized eigenvalue problems on matrix pencils (depending again on several evaluations of the considered parameterizations). This is the so-called “polynomial algebra by values” approach where the huge polynomial equations coming from Elimination Theory (e.g., using resultants) are replaced by big structured and sparse numerical matrices. For these matrices there are well-known numerical techniques allowing to provide the results we need to answer the geometric questions on the considered curves and surfaces

Total Degree Formula for the Generic Offset to a Parametric Surface

We provide a resultant-based formula for the total degree w.r.t. the spatial variables of the generic offset to a parametric surface. The parametrization of the surface is not assumed to be proper.Comment: Preprint of an article to be published at the International Journal of Algebra and Computation, World Scientific Publishing, DOI:10.1142/S021819671100680