5,742 research outputs found
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 plane curves parameterizable by conics
We introduce the class of rational plane curves parameterizable by conics as
an extension of the family of curves parameterizable by lines (also known as
monoid curves). We show that they are the image of monoid curves via suitable
quadratic transformations in projective plane. We also describe all the
possible proper parameterizations of them, and a set of minimal generators of
the Rees Algebra associated to these parameterizations, extending well-known
results for curves parameterizable by lines.Comment: 28 pages, 1 figure. Revised version. Accepted for publication in
Journal of Algebr
The geometry of some parameterizations and encodings
We explore parameterizations by radicals of low genera algebraic curves. We
prove that for a prime power that is large enough and prime to , a fixed
positive proportion of all genus 2 curves over the field with elements can
be parameterized by -radicals. This results in the existence of a
deterministic encoding into these curves when is congruent to modulo
. We extend this construction to parameterizations by -radicals for
small odd integers , and make it explicit for
Encoding points on hyperelliptic curves over finite fields in deterministic polynomial time
We present families of (hyper)elliptic curve which admit an efficient
deterministic encoding function
Edwards curves and CM curves
Edwards curves are a particular form of elliptic curves that admit a fast,
unified and complete addition law. Relations between Edwards curves and
Montgomery curves have already been described. Our work takes the view of
parameterizing elliptic curves given by their j-invariant, a problematic that
arises from using curves with complex multiplication, for instance. We add to
the catalogue the links with Kubert parameterizations of X0(2) and X0(4). We
classify CM curves that admit an Edwards or Montgomery form over a finite
field, and justify the use of isogenous curves when needed
Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants
We revisit implicitization by interpolation in order to examine its properties in the context of sparse elimination theory. Based on the computation of a superset of the implicit support, implicitization is reduced to computing the nullspace of a numeric matrix. The approach is applicable to polynomial and rational parameterizations of curves and (hyper)surfaces of any dimension, including the case of parameterizations with base points.
Our support prediction is based on sparse (or toric) resultant theory, in order to exploit the sparsity of the input and the output. Our method may yield a multiple of the implicit equation: we characterize and quantify this situation by relating the nullspace dimension to the predicted support and its geometry. In this case, we obtain more than one multiples of the implicit equation; the latter can be obtained via multivariate polynomial gcd (or factoring).
All of the above techniques extend to the case of approximate computation, thus yielding a method of sparse approximate implicitization, which is important in tackling larger problems. We discuss our publicly available Maple implementation through several examples, including the benchmark of bicubic surface.
For a novel application, we focus on computing the discriminant of a multivariate polynomial, which characterizes the existence of multiple roots and generalizes the resultant of a polynomial system.
This yields an efficient, output-sensitive algorithm for
computing the discriminant polynomial
A study of singularities on rational curves via syzygies
Consider a rational projective curve C of degree d over an algebraically
closed field k. There are n homogeneous forms g_1,...,g_n of degree d in
B=k[x,y] which parameterize C in a birational, base point free, manner. We
study the singularities of C by studying a Hilbert-Burch matrix phi for the row
vector [g_1,...,g_n]. In the "General Lemma" we use the generalized row ideals
of phi to identify the singular points on C, their multiplicities, the number
of branches at each singular point, and the multiplicity of each branch.
Let p be a singular point on the parameterized planar curve C which
corresponds to a generalized zero of phi. In the "Triple Lemma" we give a
matrix phi' whose maximal minors parameterize the closure, in projective
2-space, of the blow-up at p of C in a neighborhood of p. We apply the General
Lemma to phi' in order to learn about the singularities of C in the first
neighborhood of p. If C has even degree d=2c and the multiplicity of C at p is
equal to c, then we apply the Triple Lemma again to learn about the
singularities of C in the second neighborhood of p.
Consider rational plane curves C of even degree d=2c. We classify curves
according to the configuration of multiplicity c singularities on or infinitely
near C. There are 7 possible configurations of such singularities. We classify
the Hilbert-Burch matrix which corresponds to each configuration. The study of
multiplicity c singularities on, or infinitely near, a fixed rational plane
curve C of degree 2c is equivalent to the study of the scheme of generalized
zeros of the fixed balanced Hilbert-Burch matrix phi for a parameterization of
C.Comment: Typos corrected and minor changes made. To appear in the Memoirs of
the AM
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