2,833 research outputs found
Signature Sequence of Intersection Curve of Two Quadrics for Exact Morphological Classification
We present an efficient method for classifying the morphology of the
intersection curve of two quadrics (QSIC) in PR3, 3D real projective space;
here, the term morphology is used in a broad sense to mean the shape,
topological, and algebraic properties of a QSIC, including singularity,
reducibility, the number of connected components, and the degree of each
irreducible component, etc. There are in total 35 different QSIC morphologies
with non-degenerate quadric pencils. For each of these 35 QSIC morphologies,
through a detailed study of the eigenvalue curve and the index function jump we
establish a characterizing algebraic condition expressed in terms of the Segre
characteristics and the signature sequence of a quadric pencil. We show how to
compute a signature sequence with rational arithmetic so as to determine the
morphology of the intersection curve of any two given quadrics. Two immediate
applications of our results are the robust topological classification of QSIC
in computing B-rep surface representation in solid modeling and the derivation
of algebraic conditions for collision detection of quadric primitives
The Euclidean distance degree of smooth complex projective varieties
We obtain several formulas for the Euclidean distance degree (ED degree) of
an arbitrary nonsingular variety in projective space: in terms of Chern and
Segre classes, Milnor classes, Chern-Schwartz-MacPherson classes, and an
extremely simple formula equating the Euclidean distance degree of X with the
Euler characteristic of an open subset of X
Genus six curves, K3 surfaces, and stable pairs
A general smooth curve of genus six lies on a quintic del Pezzo surface. In
\cite{AK11}, Artebani and Kond\=o construct a birational period map for genus
six curves by taking ramified double covers of del Pezzo surfaces. The map is
not defined for special genus six curves. In this paper, we construct a smooth
Deligne-Mumford stack parametrizing certain stable
surface-curve pairs which essentially resolves this map. Moreover, we give an
explicit description of pairs in containing special curves.Comment: This is v2. Exposition has been improved due to referee comments. To
appear in IMR
Effective reconstruction of generic genus 4 curves from their theta hyperplanes
Effective reconstruction formulas of a curve from its theta hyperplanes are
known classically in genus 2 (where the theta hyperplanes are Weierstrass
points), and 3 (where, for a generic curve, the theta hyperplanes are
bitangents to a plane quartic). However, for higher genera, no formula or
algorithm are known. In this paper we give an explicit (and simple) algorithm
for computing a generic genus 4 curve from it's theta hyperplanes.Comment: no content modification to previous version; presentation
modification following referees comment
The implicit equation of a canal surface
A canal surface is an envelope of a one parameter family of spheres. In this
paper we present an efficient algorithm for computing the implicit equation of
a canal surface generated by a rational family of spheres. By using Laguerre
and Lie geometries, we relate the equation of the canal surface to the equation
of a dual variety of a certain curve in 5-dimensional projective space. We
define the \mu-basis for arbitrary dimension and give a simple algorithm for
its computation. This is then applied to the dual variety, which allows us to
deduce the implicit equations of the the dual variety, the canal surface and
any offset to the canal surface.Comment: 26 pages, to be published in Journal of Symbolic Computatio
Changing Views on Curves and Surfaces
Visual events in computer vision are studied from the perspective of
algebraic geometry. Given a sufficiently general curve or surface in 3-space,
we consider the image or contour curve that arises by projecting from a
viewpoint. Qualitative changes in that curve occur when the viewpoint crosses
the visual event surface. We examine the components of this ruled surface, and
observe that these coincide with the iterated singular loci of the coisotropic
hypersurfaces associated with the original curve or surface. We derive
formulas, due to Salmon and Petitjean, for the degrees of these surfaces, and
show how to compute exact representations for all visual event surfaces using
algebraic methods.Comment: 31 page
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