scholarly journals The $2$-Hessian and sextactic points on plane algebraic curves

2019 ◽  
Vol 125 (1) ◽  
pp. 13-38
Author(s):  
Paul Aleksander Maugesten ◽  
Torgunn Karoline Moe
Keyword(s):  
Algebraic Curve ◽  
Algebraic Curves ◽  
Rational Curves ◽  
Veronese Embedding ◽  
Plane Algebraic Curve ◽  
Special Case

In an article from 1865, Arthur Cayley claims that given a plane algebraic curve there exists an associated $2$-Hessian curve that intersects it in its sextactic points. In this paper we fix an error in Cayley's calculations and provide the correct defining polynomial for the $2$-Hessian. In addition, we present a formula for the number of sextactic points on cuspidal curves and tie this formula to the $2$-Hessian. Lastly, we consider the special case of rational curves, where the sextactic points appear as zeros of the Wronski determinant of the 2nd Veronese embedding of the curve.

About the Tasks of Descriptive Geometry With Imaginary Solutions

2015 ◽  
Vol 3 (2) ◽  
pp. 3-8 ◽  
Author(s):  
Иванов ◽  
G. Ivanov ◽  
Дмитриева ◽  
I. Dmitrieva
Keyword(s):  
Field Theory ◽  
Algebraic Curve ◽  
Complex Space ◽  
Algebraic Curves ◽  
Descriptive Geometry ◽  
Special Cases ◽  
Common Plane ◽  
Plane Algebraic Curve ◽  
Methodological Problems ◽  
Mathematical Interpretation

The article is devoted to the discussion of the scientific methodological problems of presentation tasks of descriptive geometry along with having real and imaginary solutions. Examples of such problems are given, graphics solutions who give the wrong answers. As a consequence they resulted in some the textbooks on descriptive geometry to the emergence false claims type “ the curve degenerates to a point”, “a torus is a surface of the second order”, “conical and cylindrical surfaces are a special cases of the torsoboy surface in the case of degeneration of the ribs return torsoboy the surface at the point, etc.” In the article gives a correct mathematical interpretation of imaginary solutions the tasks by considering of examples an the determine the order and class of plane algebraic curve, the isolated point touch, of the line of intersection of surfaces of the second order with a common plane of symmetry. To obtain a mathematically valid answers the conclusion about the need for a combination of graphical and analytical solutions. This approach meets the requirements of the GEF on ensure as intrasubject discussed in this publication, and so interdisciplinary competencies. The latter have a broad outlet of descriptive geometry in complex space in the theory of algebraic curves and surfaces, kremenovic transformations, field theory, etc.

scholarly journals Distinct Distances on Algebraic Curves in the Plane

2016 ◽  
Vol 26 (1) ◽  
pp. 99-117 ◽  
Author(s):  
JÁNOS PACH ◽  
FRANK DE ZEEUW
Keyword(s):  
Lower Bound ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Parallel Lines ◽  
Plane Algebraic Curve ◽  
Concentric Circles

LetSbe a set ofnpoints in${\mathbb R}^{2}$contained in an algebraic curveCof degreed. We prove that the number of distinct distances determined bySis at leastcdn4/3, unlessCcontains a line or a circle.We also prove the lower boundcd′ min{m2/3n2/3,m2,n2} for the number of distinct distances betweenmpoints on one irreducible plane algebraic curve andnpoints on another, unless the two curves are parallel lines, orthogonal lines, or concentric circles. This generalizes a result on distances between lines of Sharir, Sheffer and Solymosi in [19].

OPTIMAL REPARAMETRIZATION OF POLYNOMIAL ALGEBRAIC CURVES

2001 ◽  
Vol 11 (04) ◽  
pp. 439-453 ◽  
Author(s):  
J. RAFAEL SENDRA ◽  
CARLOS VILLARINO
Keyword(s):  
Finite Field ◽  
Characteristic Zero ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Field Extension ◽  
Effective Algorithm ◽  
Plane Algebraic Curve

In this paper, we present an algorithm for optimally parametrizing polynomial algebraic curves. Let [Formula: see text] be a polynomial plane algebraic curve given by a polynomial parametrization [Formula: see text] , where [Formula: see text] is a finite field extension of a field [Formula: see text] of characteristic zero. We prove that if [Formula: see text] is polynomial over [Formula: see text] , then Weil's descente variety associated with [Formula: see text] is surprisingly simple; it is, in fact, a line. Applying this result we are able to derive an effective algorithm to algebraically optimal reparametrize polynomial algebraic curves.

A Note Concerning the Real Inflexions of Real, Plane, Algebraic Curves

1974 ◽  
Vol 17 (3) ◽  
pp. 411-412
Author(s):  
Gareth J. Griffith
Keyword(s):  
Algebraic Curve ◽  
Double Point ◽  
Algebraic Curves ◽  
Real Plane ◽  
The Real ◽  
Plane Algebraic Curve

Theorem. “If a crunode of a real, irreducible, plane, algebraic curve changes into an acnode via the intermediary stage of a real cusp, two real inflexions are introduced in a neighborhood of the double point.”

scholarly journals Least Squares Fitting of Piecewise Algebraic Curves

2007 ◽  
Vol 2007 ◽  
pp. 1-11 ◽  
Author(s):  
Chun-Gang Zhu ◽  
Ren-Hong Wang
Keyword(s):  
Least Squares ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Scattered Data ◽  
Energy Term ◽  
Bivariate Spline ◽  
Piecewise Algebraic Curve ◽  
Piecewise Algebraic Curves ◽  
The Given

A piecewise algebraic curve is defined as the zero contour of a bivariate spline. In this paper, we present a new method for fittingC1piecewise algebraic curves of degree 2 over type-2 triangulation to the given scattered data. By simultaneously approximating points, associated normals and tangents, and points constraints, the energy term is also considered in the method. Moreover, some examples are presented.

scholarly journals Approximate Implicitization of Parametric Curves Using Cubic Algebraic Splines

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Xiaolei Zhang ◽  
Jinming Wu
Keyword(s):  
Algebraic Curves ◽  
Approximation Error ◽  
Rational Curves ◽  
Parametric Curves ◽  
Approximate Implicitization

This paper presents an algorithm to solve the approximate implicitization of planar parametric curves using cubic algebraic splines. It applies piecewise cubic algebraic curves to give a globalG2continuity approximation to planar parametric curves. Approximation error on approximate implicitization of rational curves is given. Several examples are provided to prove that the proposed method is flexible and efficient.

scholarly journals Motivic zeta functions for curve singularities

2010 ◽  
Vol 198 ◽  
pp. 47-75 ◽  
Author(s):  
J. J. Moyano-Fernández ◽  
W. A. Zúňiga-Galindo
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Local Ring ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Zeta Functions ◽  
Poincaré Series ◽  
Poincare Series ◽  
Motivic Zeta Functions ◽  
Curve Singularities

AbstractLet X be a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristic p big enough. Given a local ring Op,x at a rational singular point P of X, we attached a universal zeta function which is a rational function and admits a functional equation if Op,x is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincaré series attached to singular points of algebraic curves. In particular, for the local ring attached to a complex analytic function in two variables, our universal zeta function specializes to the generalized Poincaré series introduced by Campillo, Delgado, and Gusein-Zade.

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