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The geometry of flex tangents to a cubic curve and its parameterizations

Jean-Marc Couveignes, Jean-Gabriel Kammerer
2012 Journal of symbolic computation  
We show how the study of the geometry of the nine flex tangents to a cubic produces pseudo-parameterizations, including the ones given by Icart, Kammerer, Lercier, Renault and Farashahi, and infinitely  ...  We will use the general recipe in Section 4 and the rational curves exhibited in Section 5 to produce several pseudo-parameterizations of a plane cubic C .  ...  A smooth plane projective cubic over a field with characteristic prime to 6 has no three tangent flexes concurrent, unless its modular invariant is zero. Position with respect to conics.  ... 
doi:10.1016/j.jsc.2011.11.003 fatcat:fbfrc2ppcvburoz6dkdepak4gu

Stroke Style Transfer [article]

Sara Shaheen, Bernard Ghanem
2017 Eurographics State of the Art Reports  
We propose a novel method to transfer sketch style at the stroke level from one free-hand line drawing to another, whereby these drawings can be from different artists.  ...  It aims to transfer the style of the input sketch at the stroke level to the style encountered in sketches by other artists.  ...  Quadratic Bézier curves are enough to model conic sections and only three control points {p i } 2 i=0 are needed. A fundamental parameter of a conic section is its eccentricity.  ... 
doi:10.2312/egsh.20171013 fatcat:6louzxu455cd7pw5rayixtrrpi

Intersecting quadrics: an efficient and exact implementation

Sylvain Lazard, Luis Mariano Peñaranda, Sylvain Petitjean
2006 Computational geometry  
Unlike existing implementations, it correctly identifies and parameterizes all the connected components of the intersection in all cases, returning parameterizations with rational functions whenever such  ...  We prove upper bounds on the size of the coefficients of the output parameterizations and compare these bounds to observed values. We give other experimental results and present some examples.  ...  Acknowledgements The authors wish to acknowledge Laurent Dupont for a preliminary implementation of the parameterization algorithm in MuPAD, Guillaume Hanrot for his C implementation of Uspensky's algorithm  ... 
doi:10.1016/j.comgeo.2005.10.004 fatcat:y6clgu3265dwxhldy7bvsyvenu

Intersecting quadrics

S. Lazard, L. M. Peñaranda, S. Petitjean
2004 Proceedings of the twentieth annual symposium on Computational geometry - SCG '04  
We prove upper bounds on the size of the coefficients of the output parameterization and compare these bounds to observed values.  ...  Unlike existing implementations, it correctly identifies and parameterizes all the connected components of the intersection in all cases, returning parameterizations with rational functions whenever such  ...  (2,000 lines): parameterization of a quadric of inertia (2, 2) with bigint coefficients going through a rational point, of a cone (resp. conic), of a cone (resp. conic) with a rational point, of a pair  ... 
doi:10.1145/997817.997880 dblp:conf/compgeom/LazardPP04 fatcat:6mekaucuu5ekhod7n2ponfawme

PHYSICS BASED GEOMETRIC DESIGN

HONG QIN
1996 International journal of shape modeling  
In this paper, we s u r v ey the diversity of shape representations ranging from the primitive polynomial to the sophisticated Non-Uniform Rational B-Spline (NURBS).  ...  Geometric modeling has proved to be crucial to computer graphics and computer aided geometric design (CAGD).  ...  NURBS can be used to precisely construct conics and rational quadric patches of various types 77].  ... 
doi:10.1142/s0218654396000099 fatcat:ctnoubb7vvaubpbxuiev5dpmpe

Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems

Asher Auel, Marcello Bernardara, Michele Bolognesi
2014 Journal des Mathématiques Pures et Appliquées  
In the latter two cases, if Y is the projective line over an algebraically closed field of characteristic zero, we relate rationality questions to categorical representability of X.  ...  ---to study semiorthogonal decompositions of the bounded derived category of X.  ...  Two useful results from the algebraic theory of quadratic forms.  ... 
doi:10.1016/j.matpur.2013.11.009 fatcat:t3witv3dqnaxlbummdpflozizq

On the arithmetic of D-brane superpotentials: lines and conics on the mirror quintic

Johannes Walcher
2012 Communications in Number Theory and Physics  
Irrational invariants from D-brane superpotentials are pursued on the mirror quintic, systematically according to the degree of a representative curve.  ...  The search for all conics invariant under a residual Z 2 symmetry reduces to an algebraic problem at the limit of our computational capabilities.  ...  I thank Anca Mustaţǎ for sharing the results of [24] . I would like to thank the KITP in Santa Barbara for sunny hospitality during the tedious writing of Section 3.  ... 
doi:10.4310/cntp.2012.v6.n2.a2 fatcat:upw6gsllbrbbnm7x2phzon5vte

On the Arithmetic of D-brane Superpotentials. Lines and Conics on the Mirror Quintic [article]

Johannes Walcher
2012 arXiv   pre-print
Irrational invariants from D-brane superpotentials are pursued on the mirror quintic, systematically according to the degree of a representative curve.  ...  The search for all conics invariant under a residual Z2 symmetry reduces to an algebraic problem at the limit of our computational capabilities.  ...  The for us characteristic property, namely that the formula (6.25) clear the d 2 from the denominators, can be used to determine the a k as algebraic numbers (integers) modk 2 .  ... 
arXiv:1201.6427v1 fatcat:zm5q7wsz5zarvohe63qtdyrtnq

Monodromy of Inhomogeneous Picard-Fuchs Equations [article]

Robert A. Jefferson, Johannes Walcher
2013 arXiv   pre-print
The examples feature a variety of residue fields, both abelian and non-abelian extensions of the rationals, thereby providing a glimpse of the arithmetic D-brane landscape.  ...  We identify all lines that are invariant under at least one permutation of the homogeneous variables, and calculate the inhomogeneous Picard-Fuchs equation.  ...  All the new cycles from section 2 turn out to have residue fields that are abelian extension of Q.  ... 
arXiv:1309.0490v1 fatcat:5hts5lql3nfczohewg6se4yij4

Monodromy of inhomogeneous Picard-Fuchs equations

Robert A. Jefferson, Johannes Walcher
2014 Communications in Number Theory and Physics  
The examples feature a variety of residue fields, both abelian and non-abelian extensions of the rationals, thereby providing a glimpse of the arithmetic D-brane landscape.  ...  We identify all lines that are invariant under at least one permutation of the homogeneous variables, and calculate the inhomogeneous Picard-Fuchs equation.  ...  Acknowledgments J.W. thanks the organizers of the Workshop "Recent advances in Hodge theory," University of British Columbia, June 2013, for the invitation to present some of these results.  ... 
doi:10.4310/cntp.2014.v8.n1.a1 fatcat:47qofrrs7vgx7bv5e4ddujixj4

Cycles, derived categories, and rationality [article]

Asher Auel, Marcello Bernardara
2016 arXiv   pre-print
Our main goal is to give a sense of recent developments in the (stable) rationality problem from the point of view of unramified cohomology and 0-cycles as well as derived categories and semiorthogonal  ...  of phantom subcategories of the derived category.  ...  We are very grateful to the organizers for the invitation to lecture there and for their great hospitality.  ... 
arXiv:1612.02415v1 fatcat:oarju3clm5dnxll4ey7xc5kpey

Point counting on curves using a gonality preserving lift [article]

Wouter Castryck, Jan Tuitman
2017 arXiv   pre-print
We study the problem of lifting curves from finite fields to number fields in a genus and gonality preserving way.  ...  We then use such a lift as input to an algorithm due to the second author for computing zeta functions of curves over finite fields using p-adic cohomology.  ...  The geometric gonality γ geom of C is a classical invariant.  ... 
arXiv:1605.02162v3 fatcat:o4wdrawxgzb3xet57larxwk2vy

On Dirichlet, Poncelet and Abel problems [article]

V.P. Burskii, A.S.Zhedanov
2017 arXiv   pre-print
It is shown that a solution is non-unique if and only if a corresponding Poncelet problem for two conics has a periodic trajectory. A set of problems is proven to be equivalent to the above problem.  ...  XY-chain and biorthogonal rational functions on elliptic grids in the theory of the Padé interpolation.  ...  John has proven several useful assertions, from which we extract the following.  ... 
arXiv:0903.2531v3 fatcat:cbbs3a7ninbk3k6wutfq7q3pre

On linear sections of the spinor tenfold. I

A. G. Kuznetsov
2018 Izvestiya: Mathematics  
Using this, we define a natural quadratic line complex associated with a linear section of X. We also discuss the Hilbert schemes of linear spaces and quadrics on X and its linear sections.  ...  equipped with a non-degenerate quadratic form.  ...  Linear sections of bigger codimension In this section we discuss linear sections of codimension from 3 to 6 (i.e., of dimension from 7 to 4). 7.1. The quadratic invariant.  ... 
doi:10.1070/im8756 fatcat:m3knzw6bznbhvdetibut4mzmj4

Rationality of Fano threefolds over non-closed fields [article]

Alexander Kuznetsov, Yuri Prokhorov
2021 arXiv   pre-print
We give necessary and sufficient conditions for unirationality and rationality of Fano threefolds of geometric Picard rank-1 over an arbitrary field of zero characteristic.  ...  If, moreover, the point x is in a general position with respect to the conic corresponding to a k-point of F 2 (X), the conic gives a 2-section of this fibration, and rationality of X follows from [AHTVA19  ...  The goal of this paper is to explore the questions of rationality for smooth Fano threefolds with geometric Picard rank one over an arbitrary field k of zero characteristic.  ... 
arXiv:1911.08949v3 fatcat:iuu4pteihra6xpjxdueynp57di
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