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Models of 2-Dimensional Hyperbolic Space and Relations Among Them; Hyperbolic Length, Lines, and Distances
Models of 2-dimensional hyperbolic space and relations among them; Hyperbolic length, lines, and distances Cheng Ka Long, Hui Kam Tong 1155109623, 1155109049 Course Teacher: Prof. Yi-Jen LEE Department of Mathematics, The Chinese University of Hong Kong MATH4900E Presentation 2, 5th October 2020 Outline Upper half-plane Model (Cheng) A Model for the Hyperbolic Plane The Riemann Sphere C Poincar´eDisc Model D (Hui) Basic properties of Poincar´eDisc Model Relation between D and other models Length and distance in the upper half-plane model (Cheng) Path integrals Distance in hyperbolic geometry Measurements in the Poincar´eDisc Model (Hui) M¨obiustransformations of D Hyperbolic length and distance in D Conclusion Boundary, Length, Orientation-preserving isometries, Geodesics and Angles Reference Upper half-plane model H Introduction to Upper half-plane model - continued Hyperbolic geometry Five Postulates of Hyperbolic geometry: 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. A circle may be described with any given point as its center and any distance as its radius. 4. All right angles are congruent. 5. For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R. Some interesting facts about hyperbolic geometry 1. Rectangles don't exist in hyperbolic geometry. 2. In hyperbolic geometry, all triangles have angle sum < π 3. In hyperbolic geometry if two triangles are similar, they are congruent. -
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. -
Algebraic Geometry Michael Stoll
Introductory Geometry Course No. 100 351 Fall 2005 Second Part: Algebraic Geometry Michael Stoll Contents 1. What Is Algebraic Geometry? 2 2. Affine Spaces and Algebraic Sets 3 3. Projective Spaces and Algebraic Sets 6 4. Projective Closure and Affine Patches 9 5. Morphisms and Rational Maps 11 6. Curves — Local Properties 14 7. B´ezout’sTheorem 18 2 1. What Is Algebraic Geometry? Linear Algebra can be seen (in parts at least) as the study of systems of linear equations. In geometric terms, this can be interpreted as the study of linear (or affine) subspaces of Cn (say). Algebraic Geometry generalizes this in a natural way be looking at systems of polynomial equations. Their geometric realizations (their solution sets in Cn, say) are called algebraic varieties. Many questions one can study in various parts of mathematics lead in a natural way to (systems of) polynomial equations, to which the methods of Algebraic Geometry can be applied. Algebraic Geometry provides a translation between algebra (solutions of equations) and geometry (points on algebraic varieties). The methods are mostly algebraic, but the geometry provides the intuition. Compared to Differential Geometry, in Algebraic Geometry we consider a rather restricted class of “manifolds” — those given by polynomial equations (we can allow “singularities”, however). For example, y = cos x defines a perfectly nice differentiable curve in the plane, but not an algebraic curve. In return, we can get stronger results, for example a criterion for the existence of solutions (in the complex numbers), or statements on the number of solutions (for example when intersecting two curves), or classification results. -
More on Parallel and Perpendicular Lines
More on Parallel and Perpendicular Lines By Ma. Louise De Las Peñas et’s look at more problems on parallel and perpendicular lines. The following results, presented Lin the article, “About Slope” will be used to solve the problems. Theorem 1. Let L1 and L2 be two distinct nonvertical lines with slopes m1 and m2, respectively. Then L1 is parallel to L2 if and only if m1 = m2. Theorem 2. Let L1 and L2 be two distinct nonvertical lines with slopes m1 and m2, respectively. Then L1 is perpendicular to L2 if and only if m1m2 = -1. Example 1. The perpendicular bisector of a segment is the line which is perpendicular to the segment at its midpoint. Find equations of the three perpendicular bisectors of the sides of a triangle with vertices at A(-4,1), B(4,5) and C(4,-3). Solution: Consider the triangle with vertices A(-4,1), B(4,5) and C(4,-3), and the midpoints E, F, G of sides AB, BC, and AC, respectively. TATSULOK FirstSecond Year Year Vol. Vol. 12 No.12 No. 1a 2a e-Pages 1 In this problem, our aim is to fi nd the equations of the lines which pass through the midpoints of the sides of the triangle and at the same time are perpendicular to the sides of the triangle. Figure 1 The fi rst step is to fi nd the midpoint of every side. The next step is to fi nd the slope of the lines containing every side. The last step is to obtain the equations of the perpendicular bisectors. -
Arxiv:1910.11630V1 [Math.AG] 25 Oct 2019 3 Geometric Invariant Theory 10 3.1 Quotients and the Notion of Stability
Geometric Invariant Theory, holomorphic vector bundles and the Harder–Narasimhan filtration Alfonso Zamora Departamento de Matem´aticaAplicada y Estad´ıstica Universidad CEU San Pablo Juli´anRomea 23, 28003 Madrid, Spain e-mail: [email protected] Ronald A. Z´u˜niga-Rojas Centro de Investigaciones Matem´aticasy Metamatem´aticas CIMM Escuela de Matem´atica,Universidad de Costa Rica UCR San Jos´e11501, Costa Rica e-mail: [email protected] Abstract. This survey intends to present the basic notions of Geometric Invariant Theory (GIT) through its paradigmatic application in the construction of the moduli space of holomorphic vector bundles. Special attention is paid to the notion of stability from different points of view and to the concept of maximal unstability, represented by the Harder-Narasimhan filtration and, from which, correspondences with the GIT picture and results derived from stratifications on the moduli space are discussed. Keywords: Geometric Invariant Theory, Harder-Narasimhan filtration, moduli spaces, vector bundles, Higgs bundles, GIT stability, symplectic stability, stratifications. MSC class: 14D07, 14D20, 14H10, 14H60, 53D30 Contents 1 Introduction 2 2 Preliminaries 4 2.1 Lie groups . .4 2.2 Lie algebras . .6 2.3 Algebraic varieties . .7 2.4 Vector bundles . .8 arXiv:1910.11630v1 [math.AG] 25 Oct 2019 3 Geometric Invariant Theory 10 3.1 Quotients and the notion of stability . 10 3.2 Hilbert-Mumford criterion . 14 3.3 Symplectic stability . 18 3.4 Examples . 21 3.5 Maximal unstability . 24 2 git, hvb & hnf 4 Moduli Space of vector bundles 28 4.1 GIT construction of the moduli space . 28 4.2 Harder-Narasimhan filtration . -
Combination of Cubic and Quartic Plane Curve
IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 2 (Mar. - Apr. 2013), PP 43-53 www.iosrjournals.org Combination of Cubic and Quartic Plane Curve C.Dayanithi Research Scholar, Cmj University, Megalaya Abstract The set of complex eigenvalues of unistochastic matrices of order three forms a deltoid. A cross-section of the set of unistochastic matrices of order three forms a deltoid. The set of possible traces of unitary matrices belonging to the group SU(3) forms a deltoid. The intersection of two deltoids parametrizes a family of Complex Hadamard matrices of order six. The set of all Simson lines of given triangle, form an envelope in the shape of a deltoid. This is known as the Steiner deltoid or Steiner's hypocycloid after Jakob Steiner who described the shape and symmetry of the curve in 1856. The envelope of the area bisectors of a triangle is a deltoid (in the broader sense defined above) with vertices at the midpoints of the medians. The sides of the deltoid are arcs of hyperbolas that are asymptotic to the triangle's sides. I. Introduction Various combinations of coefficients in the above equation give rise to various important families of curves as listed below. 1. Bicorn curve 2. Klein quartic 3. Bullet-nose curve 4. Lemniscate of Bernoulli 5. Cartesian oval 6. Lemniscate of Gerono 7. Cassini oval 8. Lüroth quartic 9. Deltoid curve 10. Spiric section 11. Hippopede 12. Toric section 13. Kampyle of Eudoxus 14. Trott curve II. Bicorn curve In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a rational quartic curve defined by the equation It has two cusps and is symmetric about the y-axis. -
A Survey on the Square Peg Problem
A survey on the Square Peg Problem Benjamin Matschke∗ March 14, 2014 Abstract This is a short survey article on the 102 years old Square Peg Problem of Toeplitz, which is also called the Inscribed Square Problem. It asks whether every continuous simple closed curve in the plane contains the four vertices of a square. This article first appeared in a more compact form in Notices Amer. Math. Soc. 61(4), 2014. 1 Introduction In 1911 Otto Toeplitz [66] furmulated the following conjecture. Conjecture 1 (Square Peg Problem, Toeplitz 1911). Every continuous simple closed curve in the plane γ : S1 ! R2 contains four points that are the vertices of a square. Figure 1: Example for Conjecture1. A continuous simple closed curve in the plane is also called a Jordan curve, and it is the same as an injective map from the unit circle into the plane, or equivalently, a topological embedding S1 ,! R2. Figure 2: We do not require the square to lie fully inside γ, otherwise there are counter- examples. ∗Supported by Deutsche Telekom Stiftung, NSF Grant DMS-0635607, an EPDI-fellowship, and MPIM Bonn. This article is licensed under a Creative Commons BY-NC-SA 3.0 License. 1 In its full generality Toeplitz' problem is still open. So far it has been solved affirmatively for curves that are \smooth enough", by various authors for varying smoothness conditions, see the next section. All of these proofs are based on the fact that smooth curves inscribe generically an odd number of squares, which can be measured in several topological ways. -
Polynomial Curves and Surfaces
Polynomial Curves and Surfaces Chandrajit Bajaj and Andrew Gillette September 8, 2010 Contents 1 What is an Algebraic Curve or Surface? 2 1.1 Algebraic Curves . .3 1.2 Algebraic Surfaces . .3 2 Singularities and Extreme Points 4 2.1 Singularities and Genus . .4 2.2 Parameterizing with a Pencil of Lines . .6 2.3 Parameterizing with a Pencil of Curves . .7 2.4 Algebraic Space Curves . .8 2.5 Faithful Parameterizations . .9 3 Triangulation and Display 10 4 Polynomial and Power Basis 10 5 Power Series and Puiseux Expansions 11 5.1 Weierstrass Factorization . 11 5.2 Hensel Lifting . 11 6 Derivatives, Tangents, Curvatures 12 6.1 Curvature Computations . 12 6.1.1 Curvature Formulas . 12 6.1.2 Derivation . 13 7 Converting Between Implicit and Parametric Forms 20 7.1 Parameterization of Curves . 21 7.1.1 Parameterizing with lines . 24 7.1.2 Parameterizing with Higher Degree Curves . 26 7.1.3 Parameterization of conic, cubic plane curves . 30 7.2 Parameterization of Algebraic Space Curves . 30 7.3 Automatic Parametrization of Degree 2 Curves and Surfaces . 33 7.3.1 Conics . 34 7.3.2 Rational Fields . 36 7.4 Automatic Parametrization of Degree 3 Curves and Surfaces . 37 7.4.1 Cubics . 38 7.4.2 Cubicoids . 40 7.5 Parameterizations of Real Cubic Surfaces . 42 7.5.1 Real and Rational Points on Cubic Surfaces . 44 7.5.2 Algebraic Reduction . 45 1 7.5.3 Parameterizations without Real Skew Lines . 49 7.5.4 Classification and Straight Lines from Parametric Equations . 52 7.5.5 Parameterization of general algebraic plane curves by A-splines . -
Vector Bundles and Projective Modules
VECTOR BUNDLES AND PROJECTIVE MODULES BY RICHARD G. SWAN(i) Serre [9, §50] has shown that there is a one-to-one correspondence between algebraic vector bundles over an affine variety and finitely generated projective mo- dules over its coordinate ring. For some time, it has been assumed that a similar correspondence exists between topological vector bundles over a compact Haus- dorff space X and finitely generated projective modules over the ring of con- tinuous real-valued functions on X. A number of examples of projective modules have been given using this correspondence. However, no rigorous treatment of the correspondence seems to have been given. I will give such a treatment here and then give some of the examples which may be constructed in this way. 1. Preliminaries. Let K denote either the real numbers, complex numbers or quaternions. A X-vector bundle £ over a topological space X consists of a space F(£) (the total space), a continuous map p : E(Ç) -+ X (the projection) which is onto, and, on each fiber Fx(¡z)= p-1(x), the structure of a finite di- mensional vector space over K. These objects are required to satisfy the follow- ing condition: for each xeX, there is a neighborhood U of x, an integer n, and a homeomorphism <p:p-1(U)-> U x K" such that on each fiber <b is a X-homo- morphism. The fibers u x Kn of U x K" are X-vector spaces in the obvious way. Note that I do not require n to be a constant. -
Construction of Rational Surfaces of Degree 12 in Projective Fourspace
Construction of Rational Surfaces of Degree 12 in Projective Fourspace Hirotachi Abo Abstract The aim of this paper is to present two different constructions of smooth rational surfaces in projective fourspace with degree 12 and sectional genus 13. In particular, we establish the existences of five different families of smooth rational surfaces in projective fourspace with the prescribed invariants. 1 Introduction Hartshone conjectured that only finitely many components of the Hilbert scheme of surfaces in P4 contain smooth rational surfaces. In 1989, this conjecture was positively solved by Ellingsrud and Peskine [7]. The exact bound for the degree is, however, still open, and hence the question concerning the exact bound motivates us to find smooth rational surfaces in P4. The goal of this paper is to construct five different families of smooth rational surfaces in P4 with degree 12 and sectional genus 13. The rational surfaces in P4 were previously known up to degree 11. In this paper, we present two different constructions of smooth rational surfaces in P4 with the given invariants. Both the constructions are based on the technique of “Beilinson monad”. Let V be a finite-dimensional vector space over a field K and let W be its dual space. The basic idea behind a Beilinson monad is to represent a given coherent sheaf on P(W ) as a homology of a finite complex whose objects are direct sums of bundles of differentials. The differentials in the monad are given by homogeneous matrices over an exterior algebra E = V V . To construct a Beilinson monad for a given coherent sheaf, we typically take the following steps: Determine the type of the Beilinson monad, that is, determine each object, and then find differentials in the monad. -
Essential Concepts of Projective Geomtry
Essential Concepts of Projective Geomtry Course Notes, MA 561 Purdue University August, 1973 Corrected and supplemented, August, 1978 Reprinted and revised, 2007 Department of Mathematics University of California, Riverside 2007 Table of Contents Preface : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : i Prerequisites: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :iv Suggestions for using these notes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :v I. Synthetic and analytic geometry: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :1 1. Axioms for Euclidean geometry : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 2. Cartesian coordinate interpretations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 3 3. Lines and planes in R and R : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 II. Affine geometry : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1. Synthetic affine geometry : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 2. Affine subspaces of vector spaces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 3. Affine bases: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :19 4. Properties of coordinate -
The Trigonometry of Hyperbolic Tessellations
Canad. Math. Bull. Vol. 40 (2), 1997 pp. 158±168 THE TRIGONOMETRY OF HYPERBOLIC TESSELLATIONS H. S. M. COXETER ABSTRACT. For positive integers p and q with (p 2)(q 2) Ù 4thereis,inthe hyperbolic plane, a group [p, q] generated by re¯ections in the three sides of a triangle ABC with angles ôÛp, ôÛq, ôÛ2. Hyperbolic trigonometry shows that the side AC has length †,wherecosh†≥cÛs,c≥cos ôÛq, s ≥ sin ôÛp. For a conformal drawing inside the unit circle with centre A, we may take the sides AB and AC to run straight along radii while BC appears as an arc of a circle orthogonal to the unit circle.p The circle containing this arc is found to have radius 1Û sinh †≥sÛz,wherez≥ c2 s2, while its centre is at distance 1Û tanh †≥cÛzfrom A. In the hyperbolic triangle ABC,the altitude from AB to the right-angled vertex C is ê, where sinh ê≥z. 1. Non-Euclidean planes. The real projective plane becomes non-Euclidean when we introduce the concept of orthogonality by specializing one polarity so as to be able to declare two lines to be orthogonal when they are conjugate in this `absolute' polarity. The geometry is elliptic or hyperbolic according to the nature of the polarity. The points and lines of the elliptic plane ([11], x6.9) are conveniently represented, on a sphere of unit radius, by the pairs of antipodal points (or the diameters that join them) and the great circles (or the planes that contain them). The general right-angled triangle ABC, like such a triangle on the sphere, has ®ve `parts': its sides a, b, c and its acute angles A and B.