scholarly journals Characterizations of the $G_2(4)$ and $L_3(4)$ Near Octagons

10.37236/8476 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Bart De Bruyn
Keyword(s):  
Generalized Quadrangle ◽  
Line Geometry ◽  
Near Polygon ◽  
Generalized Polygon ◽  
Generalized Hexagon ◽  
Split Cayley Hexagon ◽  
Point Line ◽  
Line Spread

A triple $(\mathcal{S},S,\mathcal{Q})$ consisting of a near polygon $\mathcal{S}$, a line spread $S$ of $\mathcal{S}$ and a set $\mathcal{Q}$ of quads of $\mathcal{S}$ is called a polygonal triple if certain nice properties are satisfied, among which there is the requirement that the point-line geometry $\mathcal{S}'$ formed by the lines of $S$ and the quads of $\mathcal{Q}$ is itself also a near polygon. This paper addresses the problem of classifying all near polygons $\mathcal{S}$ that admit a polygonal triple $(\mathcal{S},S,\mathcal{Q})$ for which a given generalized polygon $\mathcal{S}'$ is the associated near polygon. We obtain several nonexistence results and show that the $G_2(4)$ and $L_3(4)$ near octagons are the unique near octagons that admit polygonal triples whose quads are isomorphic to the generalized quadrangle $W(2)$ and whose associated near polygons are respectively isomorphic to the dual split Cayley hexagon $H^D(4)$ and the unique generalized hexagon of order $(4,1)$.

scholarly journals Laguerre geometries and some connections to generalized quadrangles

2007 ◽  
Vol 83 (3) ◽  
pp. 335-356
Author(s):  
Matthew R. Brown
Keyword(s):  
Three Dimensional ◽  
Line Pair ◽  
Generalized Quadrangles ◽  
Constant Number ◽  
Unique Point ◽  
Line Geometry ◽  
Laguerre Geometry ◽  
Point Line ◽  
Dimensional Projective Space ◽  
Unique Circle

AbstractA Laguerre plane is a geometry of points, lines and circles where three pairwise non-collinear points lie on a unique circle, any line and circle meet uniquely and finally, given a circle C and a point Q not on it for each point P on C there is a unique circle on Q and touching C at P. We generalise to a Laguerre geometry where three pairwise non-collinear points lie on a constant number of circles. Examples and conditions on the parameters of a Laguerre geometry are given.A generalized quadrangle (GQ) is a point, line geometry in which for a non-incident point, line pair (P. m) there exists a unique point on m collinear with P. In certain cases we construct a Laguerre geometry from a GQ and conversely. Using Laguerre geometries we show that a GQ of order (s. s2) satisfying Property (G) at a pair of points is equivalent to a configuration of ovoids in three-dimensional projective space.

On point-line geometry and displacement

2004 ◽  
Vol 39 (10) ◽  
pp. 1033-1050 ◽  
Author(s):  
Yi Zhang ◽  
Kwun-Lon Ting
Keyword(s):  
Line Geometry ◽  
Point Line

scholarly journals Pseudo-embeddings of the (point, k-spaces)-geometry of PG(n, 2) and projective embeddings of DW(2n − 1, 2)

2019 ◽  
Vol 19 (1) ◽  
pp. 41-56 ◽  
Author(s):  
Bart De Bruyn
Keyword(s):  
Local Structure ◽  
Polar Space ◽  
Line Geometry ◽  
Projective Embedding ◽  
Dual Polar Space ◽  
Point Line

Abstract We classify all homogeneous pseudo-embeddings of the point-line geometry defined by the points and k-dimensional subspaces of PG(n, 2), and use this to study the local structure of homogeneous full projective embeddings of the dual polar space DW(2n − 1, 2). Our investigation allows us to distinguish n possible types for such homogeneous embeddings. For each of these n types, we construct a homogeneous full projective embedding of DW(2n − 1, 2).

scholarly journals On homomorphism-homogeneous point-line geometries

2019 ◽  
Vol 54 ◽  
pp. 101-119
Author(s):  
Ėva Jungabel
Keyword(s):  
Relational Structure ◽  
Line Geometry ◽  
Ternary Relation ◽  
Relational Structures ◽  
Uniform Hypergraphs ◽  
Line Point ◽  
Point Line ◽  
Regular Line

A relational structure is homomorphism-homogeneous if every homomorphism between finite substructures extends to an endomorphism of the structure. A point-line geometry is a non-empty set of elements called points, together with a collection of subsets, called lines, in a way that every line contains at least two points and any pair of points is contained in at most one line. A line which contains more than two points is called a regular line. Point-line geometries can alternatively be formalised as relational structures. We establish a correspondence between the point-line geometries investigated in this paper and the firstorder structures with a single ternary relation L satisfying certain axioms (i.e. that the class of point-line geometries corresponds to a subclass of 3-uniform hypergraphs). We characterise the homomorphism-homogeneous point-line geometries with two regular non-intersecting lines. Homomorphism-homogeneous pointline geometries containing two regular intersecting lines have already been classified by Masulovic.

Exact results for perturbation to total positivity and to total nonsingularity

2014 ◽  
Vol 27 ◽  
Author(s):  
Miriam Farber ◽  
Mitchell Faulk ◽  
Charles Johnson ◽  
Evan Marzion
Keyword(s):  
Nonnegative Matrix ◽  
Total Positivity ◽  
Projective Planes ◽  
Singular Matrix ◽  
Line Geometry ◽  
Positive Matrix ◽  
Totally Positive ◽  
Point Line ◽  
Totally Nonnegative Matrix ◽  
The Relationship

A study of the maximum number of equal entries in totally positive and totally nonsingular m-by-n, matrices for small values of m and n, is presented. Equal entries correspond to entries of the totally nonnegative matrix J that are not changed in producing a TP or TNS matrix. It is shown that the maximum number of equal entries in a 7-by-7 totally positive matrix is strictly smaller than that for a 7-by-7 totally non-singular matrix, but, this is the first pair (m; n) for which these maximum numbers differ. Using point-line geometry in the projective plane, a family of values for (m; n) for which these maximum numbers differ is presented. Generalization to the Hadamard core, as well as larger projective planes is also established. Finally, the relationship with C4 free graphs, along with a method for producing symmetric TP matrices with maximal symmetric arrangements of equal entries is discussed.

scholarly journals Twisted cubic and point-line incidence matrix in $$\mathrm {PG}(3,q)$$

2021 ◽  
Vol 89 (10) ◽  
pp. 2211-2233 ◽  
Author(s):  
Alexander A. Davydov ◽  
Stefano Marcugini ◽  
Fernanda Pambianco
Keyword(s):  
Incidence Matrix ◽  
Twisted Cubic ◽  
Point Line

scholarly journals A Multi-Feature Fusion Slam System Attaching Semantic Invariant to Points and Lines

Sensors ◽  
2021 ◽  
Vol 21 (4) ◽  
pp. 1196
Author(s):  
Gang Li ◽  
Yawen Zeng ◽  
Huilan Huang ◽  
Shaojian Song ◽  
Bin Liu ◽  
...  
Keyword(s):  
Feature Matching ◽  
Feature Fusion ◽  
Error Function ◽  
Line Segments ◽  
Cumulative Error ◽  
Localization And Mapping ◽  
Tracking Process ◽  
Line Features ◽  
Point Line ◽  
Segment Data

The traditional simultaneous localization and mapping (SLAM) system uses static points of the environment as features for real-time localization and mapping. When there are few available point features, the system is difficult to implement. A feasible solution is to introduce line features. In complex scenarios containing rich line segments, the description of line segments is not strongly differentiated, which can lead to incorrect association of line segment data, thus introducing errors into the system and aggravating the cumulative error of the system. To address this problem, a point-line stereo visual SLAM system incorporating semantic invariants is proposed in this paper. This system improves the accuracy of line feature matching by fusing line features with image semantic invariant information. When defining the error function, the semantic invariant is fused with the reprojection error function, and the semantic constraint is applied to reduce the cumulative error of the poses in the long-term tracking process. Experiments on the Office sequence of the TartanAir dataset and the KITTI dataset show that this system improves the matching accuracy of line features and suppresses the cumulative error of the SLAM system to some extent, and the mean relative pose error (RPE) is 1.38 and 0.0593 m, respectively.

Sign in / Sign up

Export Citation Format

Share Document