602 research outputs found
Rational Hausdorff Divisors: a New approach to the Approximate Parametrization of Curves
In this paper we introduce the notion of rational Hausdorff divisor, we
analyze the dimension and irreducibility of its associated linear system of
curves, and we prove that all irreducible real curves belonging to the linear
system are rational and are at finite Hausdorff distance among them. As a
consequence, we provide a projective linear subspace where all (irreducible)
elements are solutions to the approximate parametrization problem for a given
algebraic plane curve. Furthermore, we identify the linear system with a plane
curve that is shown to be rational and we develop algorithms to parametrize it
analyzing its fields of parametrization. Therefore, we present a generic answer
to the approximate parametrization problem. In addition, we introduce the
notion of Hausdorff curve, and we prove that every irreducible Hausdorff curve
can always be parametrized with a generic rational parametrization having
coefficients depending on as many parameters as the degree of the input curve
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
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
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
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