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Projective Geometry: a Short Introduction
Projective Geometry: A Short Introduction Lecture Notes Edmond Boyer Master MOSIG Introduction to Projective Geometry Contents 1 Introduction 2 1.1 Objective . .2 1.2 Historical Background . .3 1.3 Bibliography . .4 2 Projective Spaces 5 2.1 Definitions . .5 2.2 Properties . .8 2.3 The hyperplane at infinity . 12 3 The projective line 13 3.1 Introduction . 13 3.2 Projective transformation of P1 ................... 14 3.3 The cross-ratio . 14 4 The projective plane 17 4.1 Points and lines . 17 4.2 Line at infinity . 18 4.3 Homographies . 19 4.4 Conics . 20 4.5 Affine transformations . 22 4.6 Euclidean transformations . 22 4.7 Particular transformations . 24 4.8 Transformation hierarchy . 25 Grenoble Universities 1 Master MOSIG Introduction to Projective Geometry Chapter 1 Introduction 1.1 Objective The objective of this course is to give basic notions and intuitions on projective geometry. The interest of projective geometry arises in several visual comput- ing domains, in particular computer vision modelling and computer graphics. It provides a mathematical formalism to describe the geometry of cameras and the associated transformations, hence enabling the design of computational ap- proaches that manipulates 2D projections of 3D objects. In that respect, a fundamental aspect is the fact that objects at infinity can be represented and manipulated with projective geometry and this in contrast to the Euclidean geometry. This allows perspective deformations to be represented as projective transformations. Figure 1.1: Example of perspective deformation or 2D projective transforma- tion. Another argument is that Euclidean geometry is sometimes difficult to use in algorithms, with particular cases arising from non-generic situations (e.g. -
Robot Vision: Projective Geometry
Robot Vision: Projective Geometry Ass.Prof. Friedrich Fraundorfer SS 2018 1 Learning goals . Understand homogeneous coordinates . Understand points, line, plane parameters and interpret them geometrically . Understand point, line, plane interactions geometrically . Analytical calculations with lines, points and planes . Understand the difference between Euclidean and projective space . Understand the properties of parallel lines and planes in projective space . Understand the concept of the line and plane at infinity 2 Outline . 1D projective geometry . 2D projective geometry ▫ Homogeneous coordinates ▫ Points, Lines ▫ Duality . 3D projective geometry ▫ Points, Lines, Planes ▫ Duality ▫ Plane at infinity 3 Literature . Multiple View Geometry in Computer Vision. Richard Hartley and Andrew Zisserman. Cambridge University Press, March 2004. Mundy, J.L. and Zisserman, A., Geometric Invariance in Computer Vision, Appendix: Projective Geometry for Machine Vision, MIT Press, Cambridge, MA, 1992 . Available online: www.cs.cmu.edu/~ph/869/papers/zisser-mundy.pdf 4 Motivation – Image formation [Source: Charles Gunn] 5 Motivation – Parallel lines [Source: Flickr] 6 Motivation – Epipolar constraint X world point epipolar plane x x’ x‘TEx=0 C T C’ R 7 Euclidean geometry vs. projective geometry Definitions: . Geometry is the teaching of points, lines, planes and their relationships and properties (angles) . Geometries are defined based on invariances (what is changing if you transform a configuration of points, lines etc.) . Geometric transformations -
Binomial Partial Steiner Triple Systems Containing Complete Graphs
Graphs and Combinatorics DOI 10.1007/s00373-016-1681-3 ORIGINAL PAPER Binomial partial Steiner triple systems containing complete graphs Małgorzata Pra˙zmowska1 · Krzysztof Pra˙zmowski2 Received: 1 December 2014 / Revised: 25 January 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract We propose a new approach to studies on partial Steiner triple systems con- sisting in determining complete graphs contained in them. We establish the structure which complete graphs yield in a minimal PSTS that contains them. As a by-product we introduce the notion of a binomial PSTS as a configuration with parameters of a minimal PSTS with a complete subgraph. A representation of binomial PSTS with at least a given number of its maximal complete subgraphs is given in terms of systems of perspectives. Finally, we prove that for each admissible integer there is a binomial PSTS with this number of maximal complete subgraphs. Keywords Binomial configuration · Generalized Desargues configuration · Complete graph Mathematics Subject Classification 05B30 · 05C51 · 05B40 1 Introduction In the paper we investigate the structure which (may) yield complete graphs contained in a (partial) Steiner triple system (in short: in a PSTS). Our problem is, in fact, a particular instance of a general question, investigated in the literature, which STS’s B Małgorzata Pra˙zmowska [email protected] Krzysztof Pra˙zmowski [email protected] 1 Faculty of Mathematics and Informatics, University of Białystok, ul. Ciołkowskiego 1M, 15-245 Białystok, Poland 2 Institute of Mathematics, University of Białystok, ul. Ciołkowskiego 1M, 15-245 Białystok, Poland 123 Graphs and Combinatorics (more generally: which PSTS’s) contain/do not contain a configuration of a prescribed type. -
COMBINATORICS, Volume
http://dx.doi.org/10.1090/pspum/019 PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS Volume XIX COMBINATORICS AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island 1971 Proceedings of the Symposium in Pure Mathematics of the American Mathematical Society Held at the University of California Los Angeles, California March 21-22, 1968 Prepared by the American Mathematical Society under National Science Foundation Grant GP-8436 Edited by Theodore S. Motzkin AMS 1970 Subject Classifications Primary 05Axx, 05Bxx, 05Cxx, 10-XX, 15-XX, 50-XX Secondary 04A20, 05A05, 05A17, 05A20, 05B05, 05B15, 05B20, 05B25, 05B30, 05C15, 05C99, 06A05, 10A45, 10C05, 14-XX, 20Bxx, 20Fxx, 50A20, 55C05, 55J05, 94A20 International Standard Book Number 0-8218-1419-2 Library of Congress Catalog Number 74-153879 Copyright © 1971 by the American Mathematical Society Printed in the United States of America All rights reserved except those granted to the United States Government May not be produced in any form without permission of the publishers Leo Moser (1921-1970) was active and productive in various aspects of combin• atorics and of its applications to number theory. He was in close contact with those with whom he had common interests: we will remember his sparkling wit, the universality of his anecdotes, and his stimulating presence. This volume, much of whose content he had enjoyed and appreciated, and which contains the re• construction of a contribution by him, is dedicated to his memory. CONTENTS Preface vii Modular Forms on Noncongruence Subgroups BY A. O. L. ATKIN AND H. P. F. SWINNERTON-DYER 1 Selfconjugate Tetrahedra with Respect to the Hermitian Variety xl+xl + *l + ;cg = 0 in PG(3, 22) and a Representation of PG(3, 3) BY R. -
Collineations in Perspective
Collineations in Perspective Now that we have a decent grasp of one-dimensional projectivities, we move on to their two di- mensional analogs. Although they are more complicated, in a sense, they may be easier to grasp because of the many applications to perspective drawing. Speaking of, let's return to the triangle on the window and its shadow in its full form instead of only looking at one line. Perspective Collineation In one dimension, a perspectivity is a bijective mapping from a line to a line through a point. In two dimensions, a perspective collineation is a bijective mapping from a plane to a plane through a point. To illustrate, consider the triangle on the window plane and its shadow on the ground plane as in Figure 1. We can see that every point on the triangle on the window maps to exactly one point on the shadow, but the collineation is from the entire window plane to the entire ground plane. We understand the window plane to extend infinitely in all directions (even going through the ground), the ground also extends infinitely in all directions (we will assume that the earth is flat here), and we map every point on the window to a point on the ground. Looking at Figure 2, we see that the lamp analogy breaks down when we consider all lines through O. Although it makes sense for the base of the triangle on the window mapped to its shadow on 1 the ground (A to A0 and B to B0), what do we make of the mapping C to C0, or D to D0? C is on the window plane, underground, while C0 is on the ground. -
Can One Design a Geometry Engine? on the (Un) Decidability of Affine
Noname manuscript No. (will be inserted by the editor) Can one design a geometry engine? On the (un)decidability of certain affine Euclidean geometries Johann A. Makowsky Received: June 4, 2018/ Accepted: date Abstract We survey the status of decidabilty of the consequence relation in various ax- iomatizations of Euclidean geometry. We draw attention to a widely overlooked result by Martin Ziegler from 1980, which proves Tarski’s conjecture on the undecidability of finitely axiomatizable theories of fields. We elaborate on how to use Ziegler’s theorem to show that the consequence relations for the first order theory of the Hilbert plane and the Euclidean plane are undecidable. As new results we add: (A) The first order consequence relations for Wu’s orthogonal and metric geometries (Wen- Ts¨un Wu, 1984), and for the axiomatization of Origami geometry (J. Justin 1986, H. Huzita 1991) are undecidable. It was already known that the universal theory of Hilbert planes and Wu’s orthogonal geom- etry is decidable. We show here using elementary model theoretic tools that (B) the universal first order consequences of any geometric theory T of Pappian planes which is consistent with the analytic geometry of the reals is decidable. The techniques used were all known to experts in mathematical logic and geometry in the past but no detailed proofs are easily accessible for practitioners of symbolic computation or automated theorem proving. Keywords Euclidean Geometry · Automated Theorem Proving · Undecidability arXiv:1712.07474v3 [cs.SC] 1 Jun 2018 J.A. Makowsky Faculty of Computer Science, Technion–Israel Institute of Technology, Haifa, Israel E-mail: [email protected] 2 J.A. -
Performance Evaluation of Genetic Algorithm and Simulated Annealing in Solving Kirkman Schoolgirl Problem
FUOYE Journal of Engineering and Technology (FUOYEJET), Vol. 5, Issue 2, September 2020 ISSN: 2579-0625 (Online), 2579-0617 (Paper) Performance Evaluation of Genetic Algorithm and Simulated Annealing in solving Kirkman Schoolgirl Problem *1Christopher A. Oyeleye, 2Victoria O. Dayo-Ajayi, 3Emmanuel Abiodun and 4Alabi O. Bello 1Department of Information Systems, Ladoke Akintola University of Technology, Ogbomoso, Nigeria 2Department of Computer Science, Ladoke Akintola University of Technology, Ogbomoso, Nigeria 3Department of Computer Science, Kwara State Polytechnic, Ilorin, Nigeria 4Department of Mathematical and Physical Sciences, Afe Babalola University, Ado-Ekiti, Nigeria [email protected]|[email protected]|[email protected]|[email protected] Received: 15-FEB-2020; Reviewed: 12-MAR-2020; Accepted: 28-MAY-2020 http://dx.doi.org/10.46792/fuoyejet.v5i2.477 Abstract- This paper provides performance evaluation of Genetic Algorithm and Simulated Annealing in view of their software complexity and simulation runtime. Kirkman Schoolgirl is about arranging fifteen schoolgirls into five triplets in a week with a distinct constraint of no two schoolgirl must walk together in a week. The developed model was simulated using MATLAB version R2015a. The performance evaluation of both Genetic Algorithm (GA) and Simulated Annealing (SA) was carried out in terms of program size, program volume, program effort and the intelligent content of the program. The results obtained show that the runtime for GA and SA are 11.23sec and 6.20sec respectively. The program size for GA and SA are 2.01kb and 2.21kb, respectively. The lines of code for GA and SA are 324 and 404, respectively. The program volume for GA and SA are 1121.58 and 3127.92, respectively. -
Blocking Set Free Configurations and Their Relations to Digraphs and Hypergraphs
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector DISCRETE MATHEMATICS ELSIZI’IER Discrete Mathematics 1651166 (1997) 359.-370 Blocking set free configurations and their relations to digraphs and hypergraphs Harald Gropp* Mihlingstrasse 19. D-69121 Heidelberg, German> Abstract The current state of knowledge concerning the existence of blocking set free configurations is given together with a short history of this problem which has also been dealt with in terms of digraphs without even dicycles or 3-chromatic hypergraphs. The question is extended to the case of nonsymmetric configurations (u,, b3). It is proved that for each value of I > 3 there are only finitely many values of u for which the existence of a blocking set free configuration is still unknown. 1. Introduction In the language of configurations the existence of blocking sets was investigated in [3, 93 for the first time. Further papers [4, 201 yielded the result that the existence problem for blocking set free configurations v3 has been nearly solved. There are only 8 values of v for which it is unsettled whether there is a blocking set free configuration ~7~:u = 15,16,17,18,20,23,24,26. The existence of blocking set free configurations is equivalent to the existence of certain hypergraphs and digraphs. In these two languages results have been obtained much earlier. In Section 2 the relations between configurations, hypergraphs, and digraphs are exhibited. Furthermore, a nearly forgotten result of Steinitz is described In his dissertation of 1894 Steinitz proved the existence of a l-factor in a regular bipartite graph 20 years earlier then Kiinig. -
Finite Projective Geometries 243
FINITE PROJECTÎVEGEOMETRIES* BY OSWALD VEBLEN and W. H. BUSSEY By means of such a generalized conception of geometry as is inevitably suggested by the recent and wide-spread researches in the foundations of that science, there is given in § 1 a definition of a class of tactical configurations which includes many well known configurations as well as many new ones. In § 2 there is developed a method for the construction of these configurations which is proved to furnish all configurations that satisfy the definition. In §§ 4-8 the configurations are shown to have a geometrical theory identical in most of its general theorems with ordinary projective geometry and thus to afford a treatment of finite linear group theory analogous to the ordinary theory of collineations. In § 9 reference is made to other definitions of some of the configurations included in the class defined in § 1. § 1. Synthetic definition. By a finite projective geometry is meant a set of elements which, for sugges- tiveness, are called points, subject to the following five conditions : I. The set contains a finite number ( > 2 ) of points. It contains subsets called lines, each of which contains at least three points. II. If A and B are distinct points, there is one and only one line that contains A and B. HI. If A, B, C are non-collinear points and if a line I contains a point D of the line AB and a point E of the line BC, but does not contain A, B, or C, then the line I contains a point F of the line CA (Fig. -
Geometry: Euclid and Beyond, by Robin Hartshorne, Springer-Verlag, New York, 2000, Xi+526 Pp., $49.95, ISBN 0-387-98650-2
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 39, Number 4, Pages 563{571 S 0273-0979(02)00949-7 Article electronically published on July 9, 2002 Geometry: Euclid and beyond, by Robin Hartshorne, Springer-Verlag, New York, 2000, xi+526 pp., $49.95, ISBN 0-387-98650-2 1. Introduction The first geometers were men and women who reflected on their experiences while doing such activities as building small shelters and bridges, making pots, weaving cloth, building altars, designing decorations, or gazing into the heavens for portentous signs or navigational aides. Main aspects of geometry emerged from three strands of early human activity that seem to have occurred in most cultures: art/patterns, navigation/stargazing, and building structures. These strands developed more or less independently into varying studies and practices that eventually were woven into what we now call geometry. Art/Patterns: To produce decorations for their weaving, pottery, and other objects, early artists experimented with symmetries and repeating patterns. Later the study of symmetries of patterns led to tilings, group theory, crystallography, finite geometries, and in modern times to security codes and digital picture com- pactifications. Early artists also explored various methods of representing existing objects and living things. These explorations led to the study of perspective and then projective geometry and descriptive geometry, and (in the 20th century) to computer-aided graphics, the study of computer vision in robotics, and computer- generated movies (for example, Toy Story ). Navigation/Stargazing: For astrological, religious, agricultural, and other purposes, ancient humans attempted to understand the movement of heavenly bod- ies (stars, planets, Sun, and Moon) in the apparently hemispherical sky. -
Second Edition Volume I
Second Edition Thomas Beth Universitat¨ Karlsruhe Dieter Jungnickel Universitat¨ Augsburg Hanfried Lenz Freie Universitat¨ Berlin Volume I PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK www.cup.cam.ac.uk 40 West 20th Street, New York, NY 10011-4211, USA www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia Ruiz de Alarc´on 13, 28014 Madrid, Spain First edition c Bibliographisches Institut, Zurich, 1985 c Cambridge University Press, 1993 Second edition c Cambridge University Press, 1999 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 Printed in the United Kingdom at the University Press, Cambridge Typeset in Times Roman 10/13pt. in LATEX2ε[TB] A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Beth, Thomas, 1949– Design theory / Thomas Beth, Dieter Jungnickel, Hanfried Lenz. – 2nd ed. p. cm. Includes bibliographical references and index. ISBN 0 521 44432 2 (hardbound) 1. Combinatorial designs and configurations. I. Jungnickel, D. (Dieter), 1952– . II. Lenz, Hanfried. III. Title. QA166.25.B47 1999 5110.6 – dc21 98-29508 CIP ISBN 0 521 44432 2 hardback Contents I. Examples and basic definitions .................... 1 §1. Incidence structures and incidence matrices ............ 1 §2. Block designs and examples from affine and projective geometry ........................6 §3. t-designs, Steiner systems and configurations ......... -
Finite Projective Geometry 2Nd Year Group Project
Finite Projective Geometry 2nd year group project. B. Doyle, B. Voce, W.C Lim, C.H Lo Mathematics Department - Imperial College London Supervisor: Ambrus Pal´ June 7, 2015 Abstract The Fano plane has a strong claim on being the simplest symmetrical object with inbuilt mathematical structure in the universe. This is due to the fact that it is the smallest possible projective plane; a set of points with a subsets of lines satisfying just three axioms. We will begin by developing some theory direct from the axioms and uncovering some of the hidden (and not so hidden) symmetries of the Fano plane. Alternatively, some projective planes can be derived from vector space theory and we shall also explore this and the associated linear maps on these spaces. Finally, with the help of some theory of quadratic forms we will give a proof of the surprising Bruck-Ryser theorem, which shows that if a projective plane has order n congruent to 1 or 2 mod 4, then n is the sum of two squares. Thus we will have demonstrated fascinating links between pure mathematical disciplines by incorporating the use of linear algebra, group the- ory and number theory to explain the geometric world of projective planes. 1 Contents 1 Introduction 3 2 Basic Defintions and results 4 3 The Fano Plane 7 3.1 Isomorphism and Automorphism . 8 3.2 Ovals . 10 4 Projective Geometry with fields 12 4.1 Constructing Projective Planes from fields . 12 4.2 Order of Projective Planes over fields . 14 5 Bruck-Ryser 17 A Appendix - Rings and Fields 22 2 1 Introduction Projective planes are geometrical objects that consist of a set of elements called points and sub- sets of these elements called lines constructed following three basic axioms which give the re- sulting object a remarkable level of symmetry.