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.”

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.

scholarly journals An Algorithm for Isolating the Real Solutions of Piecewise Algebraic Curves

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Jinming Wu ◽  
Xiaolei Zhang
Keyword(s):  
Iterative Algorithm ◽  
Algebraic Curve ◽  
Spline Function ◽  
Algebraic Curves ◽  
The Real ◽  
Real Solutions ◽  
Bivariate Spline ◽  
Piecewise Algebraic Curve ◽  
Piecewise Algebraic Curves

The piecewise algebraic curve, as the set of zeros of a bivariate spline function, is a generalization of the classical algebraic curve. In this paper, an algorithm is presented to compute the real solutions of two piecewise algebraic curves. It is primarily based on the Krawczyk-Moore iterative algorithm and good initial iterative interval searching algorithm. The proposed algorithm is relatively easy to implement.

scholarly journals The number of connected components of certain real algebraic curves

2001 ◽  
Vol 25 (11) ◽  
pp. 693-701
Author(s):  
Seon-Hong Kim
Keyword(s):  
Algebraic Curve ◽  
Algebraic Curves ◽  
Connected Components ◽  
Connected Component ◽  
The Real ◽  
Real Algebraic Curves ◽  
Real Algebraic Curve

For an integern≥2, letp(z)=∏k=1n(z−αk)andq(z)=∏k=1n(z−βk), whereαk,βkare real. We find the number of connected components of the real algebraic curve{(x,y)∈ℝ2:|p(x+iy)|−|q(x+iy)|=0}for someαkandβk. Moreover, in these cases, we show that each connected component contains zeros ofp(z)+q(z), and we investigate the locus of zeros ofp(z)+q(z).

scholarly journals Examples of computation of level lines of polynomials in a plane

2021 ◽  
pp. 1-36
Author(s):  
Alexander Dmitrievich Bruno ◽  
Alexander Borisovich Batkhin ◽  
Zafar Khaydar ugli Khaydarov
Keyword(s):  
Algebraic Curve ◽  
Gröbner Basis ◽  
Real Plane ◽  
Grobner Basis ◽  
Computational Algebra ◽  
The Real ◽  
Level Lines ◽  
Real Polynomials ◽  
Basis Construction

Here we present a theory and 3 nontrivial examples of level lines calculating of real polynomials in the real plane. For this case we implement the following programs of computational algebra: factorization of a polynomial, calculation of the Grobner basis, construction of Newton's polygon, representation of an algebraic curve in a plane. Furthermore, it is shown how to overcome computational difficulties.

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.

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