Offsets, Conchoids and Pedal Surfaces

We discuss three geometric constructions and their relations, namely the offset, the conchoid and the pedal construction. The offset surface F d of a given surface F is the set of points at fixed normal distance d of F. The conchoid surface G d of a given surface G is obtained by increasing the radius function by d with respect to a given reference point O. There is a nice relation between offsets and conchoids: The pedal surfaces of a family of offset surfaces are a family of conchoid surfaces. Since this relation is birational, a family of rational offset surfaces corresponds to a family of rational conchoid surfaces and vice versa. We present theoretical principles of this mapping and apply it to ruled surfaces and quadrics. Since these surfaces have rational offsets and conchoids, their pedal and inverse pedal surfaces are new classes of rational conchoid surfaces and rational offset surfaces

The Relation Between Offset and Conchoid Constructions

The one-sided offset surface Fd of a given surface F is, roughly speaking, obtained by shifting the tangent planes of F in direction of its oriented normal vector. The conchoid surface Gd of a given surface G is roughly speaking obtained by increasing the distance of G to a fixed reference point O by d. Whereas the offset operation is well known and implemented in most CAD-software systems, the conchoid operation is less known, although already mentioned by the ancient Greeks, and recently studied by some authors. These two operations are algebraic and create new objects from given input objects. There is a surprisingly simple relation between the offset and the conchoid operation. As derived there exists a rational bijective quadratic map which transforms a given surface F and its offset surfaces Fd to a surface G and its conchoidal surface Gd, and vice versa. Geometric properties of this map are studied and illustrated at hand of some complete examples. Furthermore rational universal parameterizations for offsets and conchoid surfaces are provided

Poopćene konhoide

We adapt the classical definition of conchoids as known from the Euclidean plane to geometries that can be modeled within quadrics. Based on a construction by means of cross ratios, a generalized conchoid transformation is obtained. Basic properties of the generalized conchoid transformation are worked out. At hand of some prominent examples - line geometry and sphere geometry - the actions of these conchoid transformations are studied. Linear and also non-linear transformations are presented and relations to well-known transformations are disclosed.Prilagođavamo klasičnu definiciju konhoida iz euklidske ravnine geometrijama definiranim kvadrikama. Postiže se poopćena konhoidna transformacija koja se temelji na konstrukciji pomoću dvoomjera. Proučavaju se osnovna svojstva ovakve transformacije. Djelovanje poopćene konhoidne transformacije se proučava na nekim istaknutim primjerima kao što su pravčasta i sferna geometrija. Prikazuju se linearne i nelinearne transformacije te su opisane veze s dobro poznatim transformacijama

Conchoid surfaces of spheres

The conchoid of a surface $F$ with respect to given fixed point $O$ is roughly speaking the surface obtained by increasing the radius function with respect to $O$ by a constant. This paper studies {\it conchoid surfaces of spheres} and shows that these surfaces admit rational parameterizations. Explicit parameterizations of these surfaces are constructed using the relations to pencils of quadrics in $\R^3$ and $\R^4$. Moreover we point to remarkable geometric properties of these surfaces and their construction

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

Cissoid constructions of augmented rational ruled surfaces

J.R. Sendra is member of the Research Group ASYNACS (Ref.CT-CE2019/683)Given two real affine rational surfaces we derive a criterion for deciding the rationality of their cissoid. Furthermore, when one of the surfaces is augmented ruled and the other is either an augmented ruled or an augmented Steiner surface, we prove that the cissoid is rational. Furthermore, given rational parametrizations of the surfaces, we provide a rational parametrization of the cissoid.Ministerio de Economía y CompetitividadEuropean Regional Development Fun

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

First Steps Towards Radical Parametrization of Algebraic Surfaces

We introduce the notion of radical parametrization of a surface, and we provide algorithms to compute such type of parametrizations for families of surfaces, like: Fermat surfaces, surfaces with a high multiplicity (at least the degree minus 4) singularity, all irreducible surfaces of degree at most 5, all irreducible singular surfaces of degree 6, and surfaces containing a pencil of low-genus curves. In addition, we prove that radical parametrizations are preserved under certain type of geometric constructions that include offset and conchoids.Comment: 31 pages, 7 color figures. v2: added another case of genus

Konhoide na sferi

The construction of planar conchoids can be carried over to the Euclidean unit sphere. We study the case of conchoids of (spherical) lines and circles. Some elementary constructions of tangents and osculating circles are stil valid on the sphere. Further, we aim at the illustration and a precise description of the algebraic properties of the principal views of spherical conchoids, i.e., the conchoid’s images under orthogonal projections onto their symmetry planes.Konstrukcija ravninskih konhoida može se prenijeti na euklidsku jediničnu sferu. Promatramo slučaj konhoida generiranih sfernim pravcima i kružnicama. Neke elementarne konstrukcije tangenata i kružnica zakrivljenosti vrijede i za sferne konhoide. Nadalje, naš je cilj ilustracija i precizan opis algebarskih svojstava glavnih pogleda sfernih konhoida, tj. slika konhoida pri ortogonalnom projiciranju na njihove ravnine simetrije

Ultraquadrics associated to affine and projective automorphisms

The concept of ultraquadric has been introduced by the authors as a tool to algorithmically solve the problem of simplifying the coefficients of a given rational parametrization in K(a) (t1, ..., tn) of an algebraic variety of arbitrary dimension over a field extension K(a). In this context, previous work in the one-dimensional case has shown the importance of mastering the geometry of 1-dimensional ultraquadrics (hypercircles). In this paper we study, for the first time, the properties of some higher dimensional ultraquadrics, namely, those associated to automorphisms in the field K(a) (t1, ..., tn), defined by linear rational (with common denominator) or by polynomial (with inverse also polynomial) coordinates. We conclude, among many other observations, that ultraquadrics related to polynomial automorphisms can be characterized as varieties K-isomorphic to linear varieties, while ultraquadrics arising from projective automorphisms are isomorphic to the Segre embedding of a blowup of the projective space along an ideal and, in some general case, linearly isomorphic to a toric variety. We conclude with some further details about the real-complex, 2-dimensional case, showing, for instance, that this family of ultraquadrics can be presented as a collection of ruled surfaces described by pairs of hypercircles. © 2014, Springer-Verlag Berlin Heidelberg