Affine and Projective Geometry
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Configurations of Points and Lines
Configurations of Points and Lines "RANKO'RüNBAUM 'RADUATE3TUDIES IN-ATHEMATICS 6OLUME !MERICAN-ATHEMATICAL3OCIETY http://dx.doi.org/10.1090/gsm/103 Configurations of Points and Lines Configurations of Points and Lines Branko Grünbaum Graduate Studies in Mathematics Volume 103 American Mathematical Society Providence, Rhode Island Editorial Board David Cox (Chair) Steven G. Krantz Rafe Mazzeo Martin Scharlemann 2000 Mathematics Subject Classification. Primary 01A55, 01A60, 05–03, 05B30, 05C62, 51–03, 51A20, 51A45, 51E30, 52C30. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-103 Library of Congress Cataloging-in-Publication Data Gr¨unbaum, Branko. Configurations of points and lines / Branko Gr¨unbaum. p. cm. — (Graduate studies in mathematics ; v. 103) Includes bibliographical references and index. ISBN 978-0-8218-4308-6 (alk. paper) 1. Configurations. I. Title. QA607.G875 2009 516.15—dc22 2009000303 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. c 2009 by the American Mathematical Society. -
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 -
Study Guide for the Midterm. Topics: • Euclid • Incidence Geometry • Perspective Geometry • Neutral Geometry – Through Oct.18Th
Study guide for the midterm. Topics: • Euclid • Incidence geometry • Perspective geometry • Neutral geometry { through Oct.18th. You need to { Know definitions and examples. { Know theorems by names. { Be able to give a simple proof and justify every step. { Be able to explain those of Euclid's statements and proofs that appeared in the book, the problem set, the lecture, or the tutorial. Sources: • Class Notes • Problem Sets • Assigned reading • Notes on course website. The test. • Excerpts from Euclid will be provided if needed. • The incidence axioms will be provided if needed. • Postulates of neutral geometry will be provided if needed. • There will be at least one question that is similar or identical to a homework question. Tentative large list of terms. For each of the following terms, you should be able to write two sentences, in which you define it, state it, explain it, or discuss it. Euclid's geometry: - Euclid's definition of \right angle". - Euclid's definition of \parallel lines". - Euclid's \postulates" versus \common notions". - Euclid's working with geometric magnitudes rather than real numbers. - Congruence of segments and angles. - \All right angles are equal". - Euclid's fifth postulate. - \The whole is bigger than the part". - \Collapsing compass". - Euclid's reliance on diagrams. - SAS; SSS; ASA; AAS. - Base angles of an isosceles triangle. - Angles below the base of an isosceles triangle. - Exterior angle inequality. - Alternate interior angle theorem and its converse (we use the textbook's convention). - Vertical angle theorem. - The triangle inequality. - Pythagorean theorem. Incidence geometry: - collinear points. - concurrent lines. - Simple theorems of incidence geometry. - Interpretation/model for a theory. -
Arxiv:1609.06355V1 [Cs.CC] 20 Sep 2016 Outlaw Distributions And
Outlaw distributions and locally decodable codes Jop Bri¨et∗ Zeev Dvir† Sivakanth Gopi‡ September 22, 2016 Abstract Locally decodable codes (LDCs) are error correcting codes that allow for decoding of a single message bit using a small number of queries to a corrupted encoding. De- spite decades of study, the optimal trade-off between query complexity and codeword length is far from understood. In this work, we give a new characterization of LDCs using distributions over Boolean functions whose expectation is hard to approximate (in L∞ norm) with a small number of samples. We coin the term ‘outlaw distribu- tions’ for such distributions since they ‘defy’ the Law of Large Numbers. We show that the existence of outlaw distributions over sufficiently ‘smooth’ functions implies the existence of constant query LDCs and vice versa. We also give several candidates for outlaw distributions over smooth functions coming from finite field incidence geometry and from hypergraph (non)expanders. We also prove a useful lemma showing that (smooth) LDCs which are only required to work on average over a random message and a random message index can be turned into true LDCs at the cost of only constant factors in the parameters. 1 Introduction Error correcting codes (ECCs) solve the basic problem of communication over noisy channels. They encode a message into a codeword from which, even if the channel partially corrupts it, arXiv:1609.06355v1 [cs.CC] 20 Sep 2016 the message can later be retrieved. With one of the earliest applications of the probabilistic method, formally introduced by Erd˝os in 1947, pioneering work of Shannon [Sha48] showed the existence of optimal (capacity-achieving) ECCs. -
Connections Between Graph Theory, Additive Combinatorics, and Finite
UNIVERSITY OF CALIFORNIA, SAN DIEGO Connections between graph theory, additive combinatorics, and finite incidence geometry A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Michael Tait Committee in charge: Professor Jacques Verstra¨ete,Chair Professor Fan Chung Graham Professor Ronald Graham Professor Shachar Lovett Professor Brendon Rhoades 2016 Copyright Michael Tait, 2016 All rights reserved. The dissertation of Michael Tait is approved, and it is acceptable in quality and form for publication on microfilm and electronically: Chair University of California, San Diego 2016 iii DEDICATION To Lexi. iv TABLE OF CONTENTS Signature Page . iii Dedication . iv Table of Contents . .v List of Figures . vii Acknowledgements . viii Vita........................................x Abstract of the Dissertation . xi 1 Introduction . .1 1.1 Polarity graphs and the Tur´annumber for C4 ......2 1.2 Sidon sets and sum-product estimates . .3 1.3 Subplanes of projective planes . .4 1.4 Frequently used notation . .5 2 Quadrilateral-free graphs . .7 2.1 Introduction . .7 2.2 Preliminaries . .9 2.3 Proof of Theorem 2.1.1 and Corollary 2.1.2 . 11 2.4 Concluding remarks . 14 3 Coloring ERq ........................... 16 3.1 Introduction . 16 3.2 Proof of Theorem 3.1.7 . 21 3.3 Proof of Theorems 3.1.2 and 3.1.3 . 23 3.3.1 q a square . 24 3.3.2 q not a square . 26 3.4 Proof of Theorem 3.1.8 . 34 3.5 Concluding remarks on coloring ERq ........... 36 4 Chromatic and Independence Numbers of General Polarity Graphs . -
Chapter 7 Block Designs
Chapter 7 Block Designs One simile that solitary shines In the dry desert of a thousand lines. Epilogue to the Satires, ALEXANDER POPE The collection of all subsets of cardinality k of a set with v elements (k < v) has the ³ ´ v¡t property that any subset of t elements, with 0 · t · k, is contained in precisely k¡t subsets of size k. The subsets of size k provide therefore a nice covering for the subsets of a lesser cardinality. Observe that the number of subsets of size k that contain a subset of size t depends only on v; k; and t and not on the specific subset of size t in question. This is the essential defining feature of the structures that we wish to study. The example we just described inspires general interest in producing similar coverings ³ ´ v without using all the k subsets of size k but rather as small a number of them as possi- 1 2 CHAPTER 7. BLOCK DESIGNS ble. The coverings that result are often elegant geometrical configurations, of which the projective and affine planes are examples. These latter configurations form nice coverings only for the subsets of cardinality 2, that is, any two elements are in the same number of these special subsets of size k which we call blocks (or, in certain instances, lines). A collection of subsets of cardinality k, called blocks, with the property that every subset of size t (t · k) is contained in the same number (say ¸) of blocks is called a t-design. We supply the reader with constructions for t-designs with t as high as 5. -
Keith E. Mellinger
Keith E. Mellinger Department of Mathematics home University of Mary Washington 1412 Kenmore Avenue 1301 College Avenue, Trinkle Hall Fredericksburg, VA 22401 Fredericksburg, VA 22401-5300 (540) 368-3128 ph: (540) 654-1333, fax: (540) 654-2445 [email protected] http://www.keithmellinger.com/ Professional Experience • Director of the Quality Enhancement Plan, 2013 { present. UMW. • Professor of Mathematics, 2014 { present. Department of Mathematics, UMW. • Interim Director, Office of Academic and Career Services, Fall 2013 { Spring 2014. UMW. • Associate Professor, 2008 { 2014. Department of Mathematics, UMW. • Department Chair, 2008 { 2013. Department of Mathematics, UMW. • Assistant Professor, 2003 { 2008. Department of Mathematics, UMW. • Research Assistant Professor (Vigre post-doc), 2001 - 2003. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607-7045. Position partially funded by a VIGRE grant from the National Science Foundation. Education • Ph.D. in Mathematics: August 2001, University of Delaware, Newark, DE. { Thesis: Mixed Partitions and Spreads of Projective Spaces • M.S. in Mathematics: May 1997, University of Delaware. • B.S. in Mathematics: May 1995, Millersville University, Millersville, PA. { minor in computer science, option in statistics Grants and Awards • Millersville University Outstanding Young Alumni Achievement Award, presented by the alumni association at MU, May 2013. • Mathematical Association of America's Carl B. Allendoerfer Award for the article, -
On the Levi Graph of Point-Line Configurations 895
inv lve a journal of mathematics On the Levi graph of point-line configurations Jessica Hauschild, Jazmin Ortiz and Oscar Vega msp 2015 vol. 8, no. 5 INVOLVE 8:5 (2015) msp dx.doi.org/10.2140/involve.2015.8.893 On the Levi graph of point-line configurations Jessica Hauschild, Jazmin Ortiz and Oscar Vega (Communicated by Joseph A. Gallian) We prove that the well-covered dimension of the Levi graph of a point-line configuration with v points, b lines, r lines incident with each point, and every line containing k points is equal to 0, whenever r > 2. 1. Introduction The concept of the well-covered space of a graph was first introduced by Caro, Ellingham, Ramey, and Yuster[Caro et al. 1998; Caro and Yuster 1999] as an effort to generalize the study of well-covered graphs. Brown and Nowakowski [2005] continued the study of this object and, among other things, provided several examples of graphs featuring odd behaviors regarding their well-covered spaces. One of these special situations occurs when the well-covered space of the graph is trivial, i.e., when the graph is anti-well-covered. In this work, we prove that almost all Levi graphs of configurations in the family of the so-called .vr ; bk/- configurations (see Definition 3) are anti-well-covered. We start our exposition by providing the following definitions and previously known results. Any introductory concepts we do not present here may be found in the books by Bondy and Murty[1976] and Grünbaum[2009]. We consider only simple and undirected graphs. -
1 Definition and Models of Incidence Geometry
1 Definition and Models of Incidence Geometry (1.1) Definition (geometry) A geometry is a set S of points and a set L of lines together with relationships between the points and lines. p 2 1. Find four points which belong to set M = f(x;y) 2 R jx = 5g: 2. Let M = f(x;y)jx;y 2 R;x < 0g: p (i) Find three points which belong to set N = f(x;y) 2 Mjx = 2g. − 4 L f 2 Mj 1 g (ii) Are points P (3;3), Q(6;4) and R( 2; 3 ) belong to set = (x;y) y = 3 x + 2 ? (1.2) Definition (Cartesian Plane) L Let E be the set of all vertical and non-vertical lines, La and Lk;n where vertical lines: f 2 2 j g La = (x;y) R x = a; a is fixed real number ; non-vertical lines: f 2 2 j g Lk;n = (x;y) R y = kx + n; k and n are fixed real numbers : C f 2 L g The model = R ; E is called the Cartesian Plane. C f 2 L g 3. Let = R ; E denote Cartesian Plane. (i) Find three different points which belong to Cartesian vertical line L7. (ii) Find three different points which belong to Cartesian non-vertical line L15;p2. C f 2 L g 4. Let P be some point in Cartesian Plane = R ; E . Show that point P cannot lie simultaneously on both L and L (where a a ). a a0 , 0 (1.3) Definition (Poincar´ePlane) L Let H be the set of all type I and type II lines, aL and pLr where type I lines: f 2 j g aL = (x;y) H x = a; a is a fixed real number ; type II lines: f 2 j − 2 2 2 2 g pLr = (x;y) H (x p) + y = r ; p and r are fixed R, r > 0 ; 2 H = f(x;y) 2 R jy > 0g: H f L g The model = H; H will be called the Poincar´ePlane. -
The Dual Theorem Concerning Aubert Line
The Dual Theorem concerning Aubert Line Professor Ion Patrascu, National College "Buzeşti Brothers" Craiova - Romania Professor Florentin Smarandache, University of New Mexico, Gallup, USA In this article we introduce the concept of Bobillier transversal of a triangle with respect to a point in its plan; we prove the Aubert Theorem about the collinearity of the orthocenters in the triangles determined by the sides and the diagonals of a complete quadrilateral, and we obtain the Dual Theorem of this Theorem. Theorem 1 (E. Bobillier) Let 퐴퐵퐶 be a triangle and 푀 a point in the plane of the triangle so that the perpendiculars taken in 푀, and 푀퐴, 푀퐵, 푀퐶 respectively, intersect the sides 퐵퐶, 퐶퐴 and 퐴퐵 at 퐴푚, 퐵푚 and 퐶푚. Then the points 퐴푚, 퐵푚 and 퐶푚 are collinear. 퐴푚퐵 Proof We note that = 퐴푚퐶 aria (퐵푀퐴푚) (see Fig. 1). aria (퐶푀퐴푚) 1 Area (퐵푀퐴푚) = ∙ 퐵푀 ∙ 푀퐴푚 ∙ 2 sin(퐵푀퐴푚̂ ). 1 Area (퐶푀퐴푚) = ∙ 퐶푀 ∙ 푀퐴푚 ∙ 2 sin(퐶푀퐴푚̂ ). Since 1 3휋 푚(퐶푀퐴푚̂ ) = − 푚(퐴푀퐶̂ ), 2 it explains that sin(퐶푀퐴푚̂ ) = − cos(퐴푀퐶̂ ); 휋 sin(퐵푀퐴푚̂ ) = sin (퐴푀퐵̂ − ) = − cos(퐴푀퐵̂ ). 2 Therefore: 퐴푚퐵 푀퐵 ∙ cos(퐴푀퐵̂ ) = (1). 퐴푚퐶 푀퐶 ∙ cos(퐴푀퐶̂ ) In the same way, we find that: 퐵푚퐶 푀퐶 cos(퐵푀퐶̂ ) = ∙ (2); 퐵푚퐴 푀퐴 cos(퐴푀퐵̂ ) 퐶푚퐴 푀퐴 cos(퐴푀퐶̂ ) = ∙ (3). 퐶푚퐵 푀퐵 cos(퐵푀퐶̂ ) The relations (1), (2), (3), and the reciprocal Theorem of Menelaus lead to the collinearity of points 퐴푚, 퐵푚, 퐶푚. Note Bobillier's Theorem can be obtained – by converting the duality with respect to a circle – from the theorem relative to the concurrency of the heights of a triangle. -
The Generalization of Miquels Theorem
THE GENERALIZATION OF MIQUEL’S THEOREM ANDERSON R. VARGAS Abstract. This papper aims to present and demonstrate Clifford’s version for a generalization of Miquel’s theorem with the use of Euclidean geometry arguments only. 1. Introduction At the end of his article, Clifford [1] gives some developments that generalize the three circles version of Miquel’s theorem and he does give a synthetic proof to this generalization using arguments of projective geometry. The series of propositions given by Clifford are in the following theorem: Theorem 1.1. (i) Given three straight lines, a circle may be drawn through their intersections. (ii) Given four straight lines, the four circles so determined meet in a point. (iii) Given five straight lines, the five points so found lie on a circle. (iv) Given six straight lines, the six circles so determined meet in a point. That can keep going on indefinitely, that is, if n ≥ 2, 2n straight lines determine 2n circles all meeting in a point, and for 2n +1 straight lines the 2n +1 points so found lie on the same circle. Remark 1.2. Note that in the set of given straight lines, there is neither a pair of parallel straight lines nor a subset with three straight lines that intersect in one point. That is being considered all along the work, without further ado. arXiv:1812.04175v1 [math.HO] 11 Dec 2018 In order to prove this generalization, we are going to use some theorems proposed by Miquel [3] and some basic lemmas about a bunch of circles and their intersections, and we will follow the idea proposed by Lebesgue[2] in a proof by induction.