CROSS RATIO GRAPHS

2001 ◽  
Vol 64 (2) ◽  
pp. 257-272 ◽  
Author(s):  
A. GARDINER ◽  
CHERYL E. PRAEGER ◽  
SANMING ZHOU
Keyword(s):  
Automorphism Groups ◽  
Projective Line ◽  
Cross Ratio ◽  
Transitive Graph ◽  
Combinatorial Properties ◽  
The Family ◽  
The Cross ◽  
Transitive Graphs ◽  
Ordered Pairs

A family of arc-transitive graphs is studied. The vertices of these graphs are ordered pairs of distinct points from a finite projective line, and adjacency is defined in terms of the cross ratio. A uniform description of the graphs is given, their automorphism groups are determined, the problem of isomorphism between graphs in the family is solved, some combinatorial properties are explored, and the graphs are characterised as a certain class of arc-transitive graphs. Some of these graphs have arisen as examples in studies of arc-transitive graphs with complete quotients and arc-transitive graphs which ‘almost cover’ a 2-arc transitive graph.

Distributions of projective invariants and model-based machine vision

1996 ◽  
Vol 28 (03) ◽  
pp. 641-661 ◽  
Author(s):  
K. V. Mardia ◽  
Colin Goodall ◽  
Alistair Walder
Keyword(s):  
Machine Vision ◽  
Projective Transformation ◽  
Likelihood Estimation ◽  
Cauchy Distribution ◽  
Projective Line ◽  
Cross Ratio ◽  
Projective Invariants ◽  
Inference Problems ◽  
The Cross ◽  
Insight Into

In machine vision, objects are observed subject to an unknown projective transformation, and it is usual to use projective invariants for either testing for a false alarm or for classifying an object. For four collinear points, the cross-ratio is the simplest statistic which is invariant under projective transformations. We obtain the distribution of the cross-ratio under the Gaussian error model with different means. The case of identical means, which has appeared previously in the literature, is derived as a particular case. Various alternative forms of the cross-ratio density are obtained, e.g. under the Casey arccos transformation, and under an arctan transformation from the real projective line of cross-ratios to the unit circle. The cross-ratio distributions are novel to the probability literature; surprisingly various types of Cauchy distribution appear. To gain some analytical insight into the distribution, a simple linear-ratio is also introduced. We also give some results for the projective invariants of five coplanar points. We discuss the general moment properties of the cross-ratio, and consider some inference problems, including maximum likelihood estimation of the parameters.

scholarly journals Cross-ratio Dynamics on Ideal Polygons

2020 ◽  
Author(s):  
Maxim Arnold ◽  
Dmitry Fuchs ◽  
Ivan Izmestiev ◽  
Serge Tabachnikov
Keyword(s):  
Hyperbolic Space ◽  
Moduli Space ◽  
Hyperbolic Plane ◽  
Projective Line ◽  
Cross Ratio ◽  
Poisson Structures ◽  
Completely Integrable ◽  
The Cross

Abstract Two ideal polygons, $(p_1,\ldots ,p_n)$ and $(q_1,\ldots ,q_n)$, in the hyperbolic plane or in hyperbolic space are said to be $\alpha $-related if the cross-ratio $[p_i,p_{i+1},q_i,q_{i+1}] = \alpha $ for all $i$ (the vertices lie on the projective line, real or complex, respectively). For example, if $\alpha = -1$, the respective sides of the two polygons are orthogonal. This relation extends to twisted ideal polygons, that is, polygons with monodromy, and it descends to the moduli space of Möbius-equivalent polygons. We prove that this relation, which is generically a 2-2 map, is completely integrable in the sense of Liouville. We describe integrals and invariant Poisson structures and show that these relations, with different values of the constants $\alpha $, commute, in an appropriate sense. We investigate the case of small-gons and describe the exceptional ideal pentagons and hexagons that possess infinitely many $\alpha $-related polygons.

Distributions of projective invariants and model-based machine vision

1996 ◽  
Vol 28 (3) ◽  
pp. 641-661 ◽  
Author(s):  
K. V. Mardia ◽  
Colin Goodall ◽  
Alistair Walder
Keyword(s):  
Machine Vision ◽  
Projective Transformation ◽  
Likelihood Estimation ◽  
Cauchy Distribution ◽  
Projective Line ◽  
Cross Ratio ◽  
Projective Invariants ◽  
Inference Problems ◽  
The Cross ◽  
Insight Into

In machine vision, objects are observed subject to an unknown projective transformation, and it is usual to use projective invariants for either testing for a false alarm or for classifying an object. For four collinear points, the cross-ratio is the simplest statistic which is invariant under projective transformations. We obtain the distribution of the cross-ratio under the Gaussian error model with different means. The case of identical means, which has appeared previously in the literature, is derived as a particular case. Various alternative forms of the cross-ratio density are obtained, e.g. under the Casey arccos transformation, and under an arctan transformation from the real projective line of cross-ratios to the unit circle. The cross-ratio distributions are novel to the probability literature; surprisingly various types of Cauchy distribution appear. To gain some analytical insight into the distribution, a simple linear-ratio is also introduced. We also give some results for the projective invariants of five coplanar points. We discuss the general moment properties of the cross-ratio, and consider some inference problems, including maximum likelihood estimation of the parameters.

scholarly journals Partitions on the Projective Plane Over Galois Field of Order 11^m, m=1, 2, 3

2020 ◽  
pp. 204-214
Author(s):  
Sura M.A. Al-subahawi ◽  
Najm Abdulzahra Makhrib Al-seraji
Keyword(s):  
Finite Field ◽  
Projective Plane ◽  
Group Actions ◽  
Galois Field ◽  
Projective Line ◽  
Cross Ratio ◽  
Arc Length ◽  
The Cross

This research is concerned with the study of the projective plane over a finite field . The main purpose is finding partitions of the projective line PG( ) and the projective plane PG( ) , in addition to embedding PG(1, ) into PG( ) and PG( ) into PG( ). Clearly, the orbits of PG( ) are found, along with the cross-ratio for each orbit. As for PG( ), 13 partitions were found on PG( ) each partition being classified in terms of the degree of its arc, length, its own code, as well as its error correcting. The last main aim is to classify the group actions on PG( ).

scholarly journals Further steps on the reconstruction of convex polyominoes from orthogonal projections

2021 ◽  
Author(s):  
Paolo Dulio ◽  
Andrea Frosini ◽  
Simone Rinaldi ◽  
Lama Tarsissi ◽  
Laurent Vuillon
Keyword(s):  
Discrete Geometry ◽  
Convex Subset ◽  
Convex Sets ◽  
A Priori ◽  
Orthogonal Projections ◽  
Reconstruction Process ◽  
Combinatorial Properties ◽  
The Family ◽  
Discrete Counterpart ◽  
Interior Part

AbstractA remarkable family of discrete sets which has recently attracted the attention of the discrete geometry community is the family of convex polyominoes, that are the discrete counterpart of Euclidean convex sets, and combine the constraints of convexity and connectedness. In this paper we study the problem of their reconstruction from orthogonal projections, relying on the approach defined by Barcucci et al. (Theor Comput Sci 155(2):321–347, 1996). In particular, during the reconstruction process it may be necessary to expand a convex subset of the interior part of the polyomino, say the polyomino kernel, by adding points at specific positions of its contour, without losing its convexity. To reach this goal we consider convexity in terms of certain combinatorial properties of the boundary word encoding the polyomino. So, we first show some conditions that allow us to extend the kernel maintaining the convexity. Then, we provide examples where the addition of one or two points causes a loss of convexity, which can be restored by adding other points, whose number and positions cannot be determined a priori.

Suggested Notation for Ratios and Cross-Ratios

1912 ◽  
Vol 6 (98) ◽  
pp. 294-296
Author(s):  
Alfred Lodge
Keyword(s):  
Cross Ratio ◽  
The Cross

I wish to call attention to the value, for some purposes, ot the notation for the ratio ; and for the cross-ratio . For instance: in Menelaus’ theorem for the property of a transversal meeting the sides of a triangle ABC in the points P, Q, R, the first mentioned notation makes the property shine out very clearly The equation in the form is , which conspicuously separates the points on the transversal from the angular points of the triangle.

Reclaiming their socio-economic space in African culture : Shona Women Cross-Border Traders of Zimbabwe

2021 ◽  
Vol 2 (1) ◽  
pp. 65-82
Author(s):  
Enna Sukutai Gudhlanga
Keyword(s):  
Well Being ◽  
Global South ◽  
Customary Law ◽  
African Woman ◽  
African Culture ◽  
Economic Activities ◽  
Cross Border ◽  
Success Stories ◽  
The Family ◽  
The Cross

The advent of colonialism relegated the traditional African woman to the fringes of the family and society through codified customary law. The Shona women of Zimbabwe were some of the worst affected as they were re-defined as housewives who had to rely on their husbands for the up-keep of the family. However, in as much as globalisation has been accused of having brought some crisis on the African continent and side-lined a significant number of indigenous players, for the African woman in the global south it has brought some form of re-awakening. Globalisation seems to have re-opened the avenues for Shona women and enabled them to re-negotiate their entry back into the economic activities of the family and the public sphere. Despite the general lack of interest in the activities of women and in the strategies used by the poor for survival, it is a known fact that Shona women have become a force to reckon with in terms of cross-border trading in Zimbabwe. This research was prompted by the general hub of activity at the country's borders before the onslaught of the COVID-19 pandemic and the predominance of women traders who traverse the borders but whose activities have either not attracted enough attention to get their work recognised, or simply because they are taken for granted. Despite such strides, women in the cross-border trading business have instead garnered a certain stigma around them to the extent that the magnitude of their work is largely unrecognised. Yet elsewhere, the significance of women in informal trade is well documented. This study argues that women have not been left out in the global arena of trade. Desai (2009) acknowledges that the global economic openings in the informal sector have afforded women the opportunity to become active players in the markets of the global South. It is the aim of this research to investigate how globalisation has influenced the nature of the activities of Shona women in the cross-border trading business in Zimbabwe and their impact on the social well-being of the family and the nation’s economy at large. The research is largely qualitative in nature. Purposively selected Shona female cross-border traders at the Gulf Complex and Copacabana Market in Harare were interviewed before the COVID pandemic. The study revealed that the transnational activities of these Zimbabwean women are more wide-spread than has been anticipated. The study also revealed that women are unrecognised pillars in the economy of Zimbabwe as reflected in their success stories that have benefited Zimbabwe as a country. The study was informed by Africana Womanist theory which is embedded in African culture with special leaning on Ubuntu/ Unhu philosophy which recognises the complementary roles and partnerships of both men and women in resolving society's challenges.

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