20 research outputs found
Rational parametrization of conchoids to algebraic curves
We study the rationality of each of the components of the conchoid to an irreducible algebraic affine plane curve, excluding the trivial cases of the lines through the focus and the circle centered at the focus and radius the distance involved in the conchoid. We prove that conchoids having all their components rational can only be generated by rational curves. Moreover, we show that reducible conchoids to rational curves have always their two components rational. In addition, we prove that the rationality of the conchoid component, to a rational curve, does depend on the base curve and on the focus but not on the distance. As a consequence, we provide an algorithm that analyzes the rationality of all the components of the conchoid and, in the affirmative case, parametrizes them. The algorithm only uses a proper parametrization of the base curve and the focus and, hence, does not require the previous computation of the conchoid. As a corollary, we show that the conchoid to the irreducible conics, with conchoid-focus on the conic, are rational and we give parametrizations. In particular we parametrize the Limaçons of Pascal. We also parametrize the conchoids of Nicomedes. Finally, we show how to find the foci from where the conchoid is rational or with two rational components
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
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
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
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
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
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
Radical parametrization of algebraic curves and surfaces
Parametrization of algebraic curves and surfaces is a fundamental topic in CAGD (intersections;
offsets and conchoids; etc.) There are many results on rational parametrization, in
particular in the curve case, but the class of such objects is relatively small. If we allow root
extraction, the class of parametrizable objetcs is greatly enlarged (for example, elliptic curves
can be parametrized with one square root). We will describe the basics and the state of the
art of the problem of parametrization of curves and surfaces by radicals.Junta de Extremadura and FEDER fundsThis contribution is partially supported by the Ministerio de Econom´ıa y Competitividad under
the project MTM2011-25816-C02-01, by the Austrian Science Fund (FWF) P22766-N18, and by
Junta de Extremadura and FEDER funds. The first author is a member of the of the Research
Group asynacs (Ref. ccee2011/r34)
Conchoid surfaces of spheres
The conchoid of a surface with respect to given fixed point is
roughly speaking the surface obtained by increasing the radius function with
respect to 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 and . Moreover we point to
remarkable geometric properties of these surfaces and their construction