Planar Circle Geometries an Introduction to Moebius–, Laguerre– and Minkowski–Planes
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On Collineation Groups of Finite Planes
On collineation groups of finite planes Arrigo BONISOLI Dipartimento di Matematica Universit`adella Basilicata via N.Sauro 85 85100 Potenza (Italy) Socrates Intensive Programme Finite Geometries and Their Applications Gent, April of the WMY Please send all remarks to the author's e{mail address: [email protected] 1 Introduction From the Introduction to P. Dembowski's Finite Geometries, Springer, Berlin 1968: \ ::: An alternative approach to the study of projective planes began with a paper by BAER 1942 in which the close relationship between Desargues' theorem and the existence of central collineations was pointed out. Baer's notion of (p; L){transitivity, corresponding to this relationship, proved to be extremely fruitful. On the one hand, it provided a better understanding of coordinate structures (here SCHWAN 1919 was a forerunner); on the other hand it led eventually to the only coordinate{free, and hence geometrically satisfactory, classification of projective planes existing today, namely that by LENZ 1954 and BARLOTTI 1957. Due to deep discoveries in finite group theory the analysis of this classification has been particularly penetrating for 1 finite planes in recent years. In fact, finite groups were also applied with great success to problems not connected with (p; L){transitivity. ::: The field is influenced increasingly by problems, methods, and results in the theory of finite groups, mainly for the well known reason that the study of automorphisms \has always yielded the most powerful results" (E. Artin, Geometric Algebra, Interscience, New York 1957, p. 54). Finite{geometrical arguments can serve to prove group theoretical results, too, and it seems that the fruitful interplay between finite geometries and finite groups will become even closer in the future. -
FINITE FOURIER SERIES and OVALS in PG(2, 2H)
J. Aust. Math. Soc. 81 (2006), 21-34 FINITE FOURIER SERIES AND OVALS IN PG(2, 2h) J. CHRIS FISHERY and BERNHARD SCHMIDT (Received 2 April 2004; revised 7 May 2005) Communicated by L. Batten Abstract We propose the use of finite Fourier series as an alternative means of representing ovals in projective planes of even order. As an example to illustrate the method's potential, we show that the set [w1 + wy' + w ~y' : 0 < j < 2;'| c GF(22/1) forms an oval if w is a primitive (2h + l)sl root of unity in GF(22'') and GF(22'') is viewed as an affine plane over GF(2;'). For the verification, we only need some elementary 'trigonometric identities' and a basic irreducibility lemma that is of independent interest. Finally, we show that our example is the Payne oval when h is odd, and the Adelaide oval when h is even. 2000 Mathematics subject classification: primary 51E20, 05B25. 1. Introduction In any finite projective plane of order q, an oval is a set of q + 1 points, no three of which are collinear. In the classical plane PG(2, q) over GF(g), a nondegenerate conic is the prototypical oval. If the order of a plane is even, then the tangents to an oval all pass through a point that is called the nucleus of the oval. We call an oval together with its nucleus a hyperoval. During the 1950s Beniamino Segre proved that in PG(2, q), (1) when q is odd, then there exist no ovals other than the conies, and (2) when q — 2\ coordinates may be chosen so that the points of a hyperoval are the elements of the set {(*, f{x), 1) | JC G GF(<?)} U {(0, 1,0), (1,0, 0)}, where / is a permutation polynomial of degree at most q — 2 for which /(0) = 0, /(I) = 1, and with the additional property that for all 5 in GF(^), the function /,. -
Arcs, Ovals, and Segre's Theorem
Arcs, Ovals, and Segre's Theorem Brian Kronenthal Most recently updated on: October 6, 2012 1 Preliminary Definitions and Results Let π be a projective plane of order q. Definition 1.1. A k-arc is a set of k points in π, no three collinear. Proposition 1.2. Let K be a k-arc. Then k ≤ q + 2. Furthermore, if q is odd, then k ≤ q + 1. Proof. Let K be a k-arc, x 2 K. Since π has q + 1 lines through any point, there are exactly q + 1 lines containing x. Furthermore, since π has a unique line through any pair of points (x; y) with y 2 K n fxg, each of the k − 1 pairs corresponds to a unique line of the plane (if two pairs corresponded to the same line, we would have 3 collinear points in K, a contradiction). Since the number of such lines cannot excceed the total number of lines in π that contain x, we conclude k − 1 ≤ q + 1; thus, k ≤ q + 2. We will now prove the contrapositive of the second statement. To that end, suppose k = q + 2. Then equality holds in the above argument, and so every line L of π with x 2 L must also contain some y 2 K nfxg. Therefore, every line of π must contain either 0 or 2 points of K. Now, choose a fixed point z2 = K. Since every pair in the set S = f(x; z)jx 2 Kg determines a line, jSj = q + 2. As every line contains 0 or 2 points of K, every line intersecting K is represented by exactly two q+2 pairs of the form (x; z), x 2 K. -
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. -
Constructing the Tits Ovoid from an Elliptic Quadric
Constructing the Tits Ovoid from an Elliptic Quadric Bill Cherowitzo UCDHSC-DDC July 1, 2006 Combinatorics 2006 Ovoids An ovoid in PG(3,q) is a set of q2+1 points, no three of which are collinear. ● Ovoids can only exist in a 3-dimensional space. ● At each point of an ovoid, the tangent lines of the ovoid through that point lie in a plane, called the tangent plane to the ovoid at that point. ● All planes in PG(3,q) intersect an ovoid either in a point (the tangent planes) or in a set of q+1 points (the secant planes). Ovoids Tangent Plane Secant Plane P The intersection of a secant plane and an ovoid (a section of the ovoid) is an oval of the secant plane (a set of q+1 points in a plane of order q no three of which are collinear.) Ovoids There are only two known families of ovoids (and no sporadic examples): Elliptic Quadrics (∃ ∀ q) Tits Ovoids (∃ iff q = 22e+1) Elliptic Quadrics An elliptic quadric in PG(3,q) is the set of points whose homogeneous coordinates satisfy a homogeneous quadratic equation and such that the set contains no line. In terms of coordinates (x,y,z,w) of PG(3,q): ● if q is odd, the set of points satisfying zw = x2 + y2 would form an elliptic quadric, but this would not work for q even since the set would include lines in that case. ● for any q, the set of points given by: {(x, y, x2 + xy + ay2, 1): x,y ∈ GF(q)} ∪ {(0,0,1,0)} where t2 + t + a is irreducible over GF(q), is an elliptic quadric. -
On Arcs in a Finite Projective Plane
ON ARCS IN A FINITE PROJECTIVE PLANE G. E. MARTIN 1. Introduction and summary. The aim of this paper is to generalize and unify results of B. Qvist, B. Segre, M. See, and others concerning arcs in a finite projective plane. The method consists of applying completely elementary combinatorial arguments. To the usual axioms for a projective plane we add the condition that the number of points be finite. Thus there exists an integer n > 2, called the order of the plane, such that the number of points and the number of lines equal n2 + n + 1 and the number of points on a line and the number of lines through a point equal n + 1. In the following, n will always denote the order of a finite plane. Desarguesian planes of order n, formed by the analytic geometry with coefficients from the Galois field of order n, are examples of finite projective planes. We shall not assume that our planes are Desarguesian, however. A set K of k > 3 points in a finite projective plane such that no three are collinear will be called a k-arc. A line containing exactly one point of K will be called a tangent of K, a line containing two points of K will be called a secant of K, and a line that does not contain any points of K will be called an exterior line of K. An (n + l)-arc in a plane of order n is called an oval. We define / by the equation n + 2 = k + t. Thus t = (n + 1) — (k — 1) is the number of tangents of K at any point of K. -
Dissertation Hyperovals, Laguerre Planes And
DISSERTATION HYPEROVALS, LAGUERRE PLANES AND HEMISYSTEMS { AN APPROACH VIA SYMMETRY Submitted by Luke Bayens Department of Mathematics In partial fulfillment of the requirements For the Degree of Doctor of Philosophy Colorado State University Fort Collins, Colorado Spring 2013 Doctoral Committee: Advisor: Tim Penttila Jeff Achter Willem Bohm Chris Peterson ABSTRACT HYPEROVALS, LAGUERRE PLANES AND HEMISYSTEMS { AN APPROACH VIA SYMMETRY In 1872, Felix Klein proposed the idea that geometry was best thought of as the study of invariants of a group of transformations. This had a profound effect on the study of geometry, eventually elevating symmetry to a central role. This thesis embodies the spirit of Klein's Erlangen program in the modern context of finite geometries { we employ knowledge about finite classical groups to solve long-standing problems in the area. We first look at hyperovals in finite Desarguesian projective planes. In the last 25 years a number of infinite families have been constructed. The area has seen a lot of activity, motivated by links with flocks, generalized quadrangles, and Laguerre planes, amongst others. An important element in the study of hyperovals and their related objects has been the determination of their groups { indeed often the only way of distinguishing them has been via such a calculation. We compute the automorphism group of the family of ovals constructed by Cherowitzo in 1998, and also obtain general results about groups acting on hyperovals, including a classification of hyperovals with large automorphism groups. We then turn our attention to finite Laguerre planes. We characterize the Miquelian Laguerre planes as those admitting a group containing a non-trivial elation and acting tran- sitively on flags, with an additional hypothesis { a quasiprimitive action on circles for planes of odd order, and insolubility of the group for planes of even order. -
Finite Projective Geometries and Linear Codes
Finite Projective Geometries and Linear Codes Tom Edgar Advisor: Dr. Anton Betten Department of Mathematics In partial fulfillment of the requirements for the degree of Master Science Fort Collins, Colorado Spring 2004 Abstract In this paper, we study the connections between linear codes and projective geometries over finite fields. Often good codes come from interesting structures in projective geometries. For example, MDS codes come from arcs (i.e. sets of points which are extremal in the sense that they admit no other than the obvious dependencies). In addition, we take a closer look at ovals and hyperovals in projective planes and ovoids in projective 3-spaces. In particular, we examine Glynn’s condition for the existence of hyperovals. Contents 1 Introduction 3 2 Projective Geometry 3 2.1 A Brief Introduction and Basic Definitions . 3 2.1.1 Quadratic Forms: Classification . 6 2.2 Arcs,Caps,andNormalRationalCurves . 8 2.3 Ovals and Hyperovals in PG2(q)................. 9 2.3.1 Glynn’s Condition for Existence of Hyperovals . 14 2.4 k-arcs, k-caps and minihypers in PGn(q)............ 25 2.4.1 k-arcsingeneral ..................... 25 2.4.2 A Glimpse into the World of k-caps: Ovoids . 29 2.4.3 Minihypers ........................ 34 3 Coding Theory 35 3.1 BasicDefinitions ......................... 35 3.2 MDSCodes ............................ 39 4 From Geometry to Linear Codes 42 4.1 Results: ArcstoMDSCodes. 42 4.1.1 Minihypers and the Griesmer Bound . 44 4.2 Conjectures ............................ 45 5 Conclusion 46 2 1 Introduction When coding theory was first introduced, the connections to projective ge- ometry were not known and hence not studied. -
Isoptics of a Closed Strictly Convex Curve. - II Rendiconti Del Seminario Matematico Della Università Di Padova, Tome 96 (1996), P
RENDICONTI del SEMINARIO MATEMATICO della UNIVERSITÀ DI PADOVA W. CIESLAK´ A. MIERNOWSKI W. MOZGAWA Isoptics of a closed strictly convex curve. - II Rendiconti del Seminario Matematico della Università di Padova, tome 96 (1996), p. 37-49 <http://www.numdam.org/item?id=RSMUP_1996__96__37_0> © Rendiconti del Seminario Matematico della Università di Padova, 1996, tous droits réservés. L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Isoptics of a Closed Strictly Convex Curve. - II. W. CIE015BLAK (*) - A. MIERNOWSKI (**) - W. MOZGAWA(**) 1. - Introduction. ’ This article is concerned with some geometric properties of isoptics which complete and deepen the results obtained in our earlier paper [3]. We therefore begin by recalling the basic notions and necessary results concerning isoptics. An a-isoptic Ca of a plane, closed, convex curve C consists of those points in the plane from which the curve is seen under the fixed angle .7r - a. We shall denote by C the set of all plane, closed, strictly convex curves. Choose an element C and a coordinate system with the ori- gin 0 in the interior of C. Let p(t), t E [ 0, 2~c], denote the support func- tion of the curve C. -
Ellipse and Oval in Baroque Sacral Architecture in Slovakia
Vol. 13, Issue 1/2017, 30-41, DOI: 10.1515/cee-2017-0004 ELLIPSE AND OVAL IN BAROQUE SACRAL ARCHITECTURE IN SLOVAKIA Zuzana GRÚ ŇOVÁ 1* , Michaela HOLEŠOVÁ 2 1 Department of Building Engineering and Urban Planning, Faculty of Civil Engineering, University of Žilina, Univerzitná 8215/1, 010 26 Žilina, Slovakia. 2 Department of Structural Mechanics and Applied Mathematics, Faculty of Civil Engineering, University of Žilina, Univerzitná 8215/1, 010 26 Žilina, Slovakia. * corresponding author: [email protected]. Abstract Keywords: Oval, circular and elliptic forms appear in the architecture from the Elliptic and oval plans; very beginning. The basic problem of the geometric analysis of the Slovak baroque sacral spaces with an elliptic or oval ground plan is a great sensitivity of the architecture; outcome calculations to the plan's precision, mainly to distinguish Geometrical analysis of Serlio's between oval and ellipse. Sebastiano Serlio and Guarino Guarini and Guarini's ovals. belong to those architects, theoreticians, who analysed the potential of circular or oval forms and some of their ideas are analysed in the paper. Elliptical or oval plans were used also in Slovak baroque architecture or interior elements and the paper introduce some of the most known examples as a connection to the world architecture ideas. 1. Introduction Oval and elliptic forms belonged to architectural dictionary of forms from the very beginning. The basic shape, easy to construct was of course a circle. Circular or many forms of circular-like, not precisely shaped forms could be found at many Neolithic sites like Skara Brae (profane and supposed sacral spaces), temples of Malta and many others. -
Types of Coordinate Systems What Are Map Projections?
What are map projections? Page 1 of 155 What are map projections? ArcGIS 10 Within ArcGIS, every dataset has a coordinate system, which is used to integrate it with other geographic data layers within a common coordinate framework such as a map. Coordinate systems enable you to integrate datasets within maps as well as to perform various integrated analytical operations such as overlaying data layers from disparate sources and coordinate systems. What is a coordinate system? Coordinate systems enable geographic datasets to use common locations for integration. A coordinate system is a reference system used to represent the locations of geographic features, imagery, and observations such as GPS locations within a common geographic framework. Each coordinate system is defined by: Its measurement framework which is either geographic (in which spherical coordinates are measured from the earth's center) or planimetric (in which the earth's coordinates are projected onto a two-dimensional planar surface). Unit of measurement (typically feet or meters for projected coordinate systems or decimal degrees for latitude–longitude). The definition of the map projection for projected coordinate systems. Other measurement system properties such as a spheroid of reference, a datum, and projection parameters like one or more standard parallels, a central meridian, and possible shifts in the x- and y-directions. Types of coordinate systems There are two common types of coordinate systems used in GIS: A global or spherical coordinate system such as latitude–longitude. These are often referred to file://C:\Documents and Settings\lisac\Local Settings\Temp\~hhB2DA.htm 10/4/2010 What are map projections? Page 2 of 155 as geographic coordinate systems. -
The Shape and History of the Ellipse in Washington, D.C
The Shape and History of the Ellipse in Washington, D.C. Clark Kimberling Department of Mathematics University of Evansville 1800 Lincoln Avenue, Evansville, IN 47722 email: [email protected] 1 Introduction When conic sections are introduced to mathematics classes, certain real-world examples are often cited. Favorites include lamp-shade shadows for hyperbolas, paths of baseballs for parabolas, and planetary orbits for ellipses. There is, however, another outstanding example of an ellipse. Known simply as the Ellipse, it is a gathering place for thousands of Americans every year, and it is probably the world’s largest noncircular ellipse. Situated just south of the White House in President’sPark, the Ellipse has an interesting shape and an interesting history. Figure 1. Looking north from the top of Washington Monument: the Ellipse and the White House [19] When Charles L’Enfant submitted his Plan for the American capital city to President George Washington, he included many squares, circles, and triangles. Today, well known shapes in or near the capital include the Federal Triangle, McPherson Square, the Pentagon, the Octagon House Museum, Washington Circle, and, of course, the Ellipse. Regarding the Ellipse in its present size and shape, a map dated September 29, 1877 (Figure 7) was probably used for the layout. The common gardener’smethod would not have been practical for so large an ellipse –nearly 17 acres –so that the question, "How was the Ellipse laid out?" is of considerable interest. (The gardener’s method uses three stakes and a rope. Drive two stakes into the ground, and let 2c be the distance between them.