scholarly journals Nets of Lines with the Combinatorics of the Square Grid and with Touching Inscribed Conics

2021 ◽  
Author(s):  
Alexander I. Bobenko ◽  
Alexander Y. Fairley
Keyword(s):  
Projective Plane ◽  
Parameter Family ◽  
Common Edge ◽  
Square Grid ◽  
Polygonal Chains ◽  
Straight Lines ◽  
Special Case

AbstractIn the projective plane, we consider congruences of straight lines with the combinatorics of the square grid and with all elementary quadrilaterals possessing touching inscribed conics. The inscribed conics of two combinatorially neighbouring quadrilaterals have the same touching point on their common edge-line. We suggest that these nets are a natural projective generalisation of incircular nets. It is shown that these nets are planar Koenigs nets. Moreover, we show that general Koenigs nets are characterised by the existence of a 1-parameter family of touching inscribed conics. It is shown that the lines of any grid of quadrilaterals with touching inscribed conics are tangent to a common conic. These grids can be constructed via polygonal chains that are inscribed in conics. The special case of billiards in conics corresponds to incircular nets.

A Combinatorial Distinction Between Unit Circles and Straight Lines: How Many Coincidences Can they Have?

2009 ◽  
Vol 18 (5) ◽  
pp. 691-705 ◽  
Author(s):  
GYÖRGY ELEKES ◽  
MIKLÓS SIMONOVITS ◽  
ENDRE SZABÓ
Keyword(s):  
Parameter Family ◽  
Sufficient Condition ◽  
Simple Application ◽  
Quadratic Number ◽  
General Sufficient Condition ◽  
Family Of Curves ◽  
Unit Circles ◽  
Triple Points ◽  
Straight Lines ◽  
Special Case

We give a very general sufficient condition for a one-parameter family of curves not to have n members with ‘too many’ (i.e., a near-quadratic number of) triple points of intersections. As a special case, a combinatorial distinction between straight lines and unit circles will be shown. (Actually, this is more than just a simple application; originally this motivated our results.)

scholarly journals A Uniqueness Result for a Simple Superlinear Eigenvalue Problem

2021 ◽  
Vol 31 (2) ◽  
Author(s):  
Michael Herrmann ◽  
Karsten Matthies
Keyword(s):  
Eigenvalue Problem ◽  
Numerical Simulations ◽  
Convolution Operator ◽  
Existence And Uniqueness ◽  
Uniqueness Result ◽  
Constitutive Laws ◽  
Parameter Family ◽  
Shape Constraint ◽  
Special Case ◽  
Nonlinear Eigenfunctions

AbstractWe study the eigenvalue problem for a superlinear convolution operator in the special case of bilinear constitutive laws and establish the existence and uniqueness of a one-parameter family of nonlinear eigenfunctions under a topological shape constraint. Our proof uses a nonlinear change of scalar parameters and applies Krein–Rutman arguments to a linear substitute problem. We also present numerical simulations and discuss the asymptotics of two limiting cases.

An Easy Proof for Some Classical Theorems in Plane Geometry

1992 ◽  
Vol 35 (4) ◽  
pp. 560-568 ◽  
Author(s):  
C. Thas
Keyword(s):  
Projective Plane ◽  
Projective Geometry ◽  
The Other ◽  
Plane Geometry ◽  
The Real ◽  
Easy Proof ◽  
Special Cases ◽  
Euclidean Planes ◽  
Straight Lines ◽  
Short Method

AbstractThe main result of this paper is a theorem about three conies in the complex or the real complexified projective plane. Is this theorem new? We have never seen it anywhere before. But since the golden age of projective geometry so much has been published about conies that it is unlikely that no one noticed this result. On the other hand, why does it not appear in the literature? Anyway, it seems interesting to "repeat" this property, because several theorems in connection with straight lines and (or) conies in projective, affine or euclidean planes are in fact special cases of this theorem. We give a few classical examples: the theorems of Pappus-Pascal, Desargues, Pascal (or its converse), the Brocard points, the point of Miquel. Finally, we have never seen in the literature a proof of these theorems using the same short method see the proof of the main theorem).

Projective Geometry in the One-Dimensional Affine Group

1964 ◽  
Vol 16 ◽  
pp. 683-700 ◽  
Author(s):  
Hans Schwerdtfeger
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Simple Group ◽  
Final Section ◽  
Theoretical Interpretation ◽  
Special Group ◽  
One Dimensional ◽  
The One ◽  
Straight Lines ◽  
Geometrical Investigation

The idea of considering the set of the elements of a group as a space, provided with a topology, measure, or metric, connected somehow with the group operation, has been used often in the work of E. Cartan and others. In the present paper we shall study a very special group whose space can be embedded naturally into a projective plane and where the straight lines have a very simple group-theoretical interpretation. It remains to be seen whether this geometrical embedding in a projective space can be extended to other classes of groups and whether the method could become an instrument of geometrical investigation, like co-ordinates or reflections. In the final section it is shown how a geometrical theorem may lead to relations within the group.

scholarly journals Plains in a Special Case. The Relationship Between the Planes

2021 ◽  
Vol 9 (11) ◽  
pp. 548-550
Author(s):  
Qozaqova Munojat Sharifjanovna
Keyword(s):  
Horizontal Projection ◽  
Straight Lines ◽  
The Given ◽  
The Relationship ◽  
Special Case

Annotation: To develop students' understanding of straight lines and planes and to develop skills and competencies in working on related issues. The listener must complete the given task on A4 paper with the necessary tools. Keywords: straight lines, perpendicular, horizontal projection, frontal projections.

Complexes of elliptic cylinders with a characteristic manifold of the generator element in the form of coordinate straight lines

2019 ◽  
pp. 82-87
Author(s):  
M. Kretov
Keyword(s):  
Three Dimensional ◽  
Parameter Family ◽  
Coordinate Plane ◽  
Coordinate Line ◽  
The Third ◽  
Characteristic Manifold ◽  
Coordinate Vector ◽  
Line Congruence ◽  
Focal Variety ◽  
Straight Lines

The complex (three-parameter family) of elliptic cylinders is investigated in the three-dimensional affine space, in which the characteristic multiplicity of the forming element consists of three coordinate axes. The focal variety of the forming element of the considered variety is geometrically characterized. Geometric properties of the complex under study were obtained. It is shown that the studied manifold exists and is determined by a completely integrable system of differential equations. It is proved that the focal variety of the forming element of the complex consists of four geometrically characterized points. The center of the ray of the straight-line congruence of the axes of the cylinder, the indicatrix of the second coordinate vector, the second coordinate line and one of the coordinate planes are fixed. The indicatrix of the first coordinate vector describes a one-parameter family of lines with tangents parallel to the second coordinate vector. The end of the first coordinate vector describes a one-parameter family of lines with tangents parallel to the third coordinate vector. The indicatrix of the third coordinate vector and its end describe congruences of planes parallel to the first coordinate plane. The points of the first coordinate line and the first coordinate plane describe one-parameter families of planes parallel to the coordinate plane indicated above.

scholarly journals Pascal’s Theorem in Real Projective Plane

2017 ◽  
Vol 25 (2) ◽  
pp. 107-119
Author(s):  
Roland Coghetto
Keyword(s):  
Projective Plane ◽  
Projective Geometry ◽  
Real Projective Plane ◽  
The Real ◽  
Hol Light ◽  
System 2 ◽  
Special Case

Summary In this article we check, with the Mizar system [2], Pascal’s theorem in the real projective plane (in projective geometry Pascal’s theorem is also known as the Hexagrammum Mysticum Theorem)1. Pappus’ theorem is a special case of a degenerate conic of two lines. For proving Pascal’s theorem, we use the techniques developed in the section “Projective Proofs of Pappus’ Theorem” in the chapter “Pappus’ Theorem: Nine proofs and three variations” [11]. We also follow some ideas from Harrison’s work. With HOL Light, he has the proof of Pascal’s theorem2. For a lemma, we use PROVER93 and OTT2MIZ by Josef Urban4 [12, 6, 7]. We note, that we don’t use Skolem/Herbrand functions (see “Skolemization” in [1]).

Conical Differentiation

1964 ◽  
Vol 16 ◽  
pp. 169-190 ◽  
Author(s):  
N. D. Lane ◽  
K. D. Singh
Keyword(s):  
Projective Plane ◽  
Affine Plane ◽  
Parameter Family ◽  
Real Projective Plane ◽  
The Real ◽  
Parabolic Case ◽  
Real Affine

This paper follows naturally a note on parabolic differentiation (2) in which the parabolically differentiable points in the real affine plane were discussed. In the parabolic case, the four-parameter family of parabolas in the affine plane led to three differentiability conditions. In the present paper, the five-parameter family of conies in the real projective plane gives rise to four differentiability conditions and a point of an arc in the projective plane is called conically differentiable if these four conditions are satisfied. The differentiable points are classified by the nature of their families of osculating conies, superosculating conies, and their ultraosculating conies.

GLOBAL CONVERGENCE OF A SPECIAL CASE OF THE DAI–YUAN FAMILY WITHOUT LINE SEARCH

2008 ◽  
Vol 25 (03) ◽  
pp. 411-420 ◽  
Author(s):  
HUI ZHU ◽  
XIONGDA CHEN
Keyword(s):  
Global Convergence ◽  
Conjugate Gradient ◽  
Line Search ◽  
Gradient Methods ◽  
Parameter Family ◽  
Conjugate Gradient Methods ◽  
Objective Functions ◽  
Nonlinear Conjugate Gradient ◽  
Nonlinear Conjugate Gradient Methods ◽  
Special Case

Conjugate gradient methods are efficient to minimize differentiable objective functions in large dimension spaces. Recently, Dai and Yuan introduced a tree-parameter family of nonlinear conjugate gradient methods and show their convergence. However, line search strategies usually bring computational burden. To overcome this problem, in this paper, we study the global convergence of a special case of three-parameter family(the CD-DY family) in which the line search procedures are replaced by fixed formulae of stepsize.

Non-Desarguesian Projective Plane Geometries Which Satisfy The Harmonic Point Axiom

1956 ◽  
Vol 8 ◽  
pp. 532-562 ◽  
Author(s):  
N. S. Mendelsohn
Keyword(s):  
Projective Plane ◽  
Desarguesian Projective Plane ◽  
Fourth Harmonic ◽  
Straight Lines

1. Introduction and summary. In her papers (12) and (13) R. Moufang discusses projective plane geometries which satisfy the axiom of the uniqueness of the fourth harmonic point. Her main result is that in such geometries non-homogeneous co-ordinates may be assigned to the points of the plane (except for the “line at infinity”) in such a way that straight lines have equations of the forms aαx + y + β = 0, or x + γ − 0.

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