Math 104: Course Summary
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Projective Coordinates and Compactification in Elliptic, Parabolic and Hyperbolic 2-D Geometry
PROJECTIVE COORDINATES AND COMPACTIFICATION IN ELLIPTIC, PARABOLIC AND HYPERBOLIC 2-D GEOMETRY Debapriya Biswas Department of Mathematics, Indian Institute of Technology-Kharagpur, Kharagpur-721302, India. e-mail: d [email protected] Abstract. A result that the upper half plane is not preserved in the hyperbolic case, has implications in physics, geometry and analysis. We discuss in details the introduction of projective coordinates for the EPH cases. We also introduce appropriate compactification for all the three EPH cases, which results in a sphere in the elliptic case, a cylinder in the parabolic case and a crosscap in the hyperbolic case. Key words. EPH Cases, Projective Coordinates, Compactification, M¨obius transformation, Clifford algebra, Lie group. 2010(AMS) Mathematics Subject Classification: Primary 30G35, Secondary 22E46. 1. Introduction Geometry is an essential branch of Mathematics. It deals with the proper- ties of figures in a plane or in space [7]. Perhaps, the most influencial book of all time, is ‘Euclids Elements’, written in around 3000 BC [14]. Since Euclid, geometry usually meant the geometry of Euclidean space of two-dimensions (2-D, Plane geometry) and three-dimensions (3-D, Solid geometry). A close scrutiny of the basis for the traditional Euclidean geometry, had revealed the independence of the parallel axiom from the others and consequently, non- Euclidean geometry was born and in projective geometry, new “points” (i.e., points at infinity and points with complex number coordinates) were intro- duced [1, 3, 9, 26]. In the eighteenth century, under the influence of Steiner, Von Staudt, Chasles and others, projective geometry became one of the chief subjects of mathematical research [7]. -
Hyperbolic Geometry
Hyperbolic Geometry David Gu Yau Mathematics Science Center Tsinghua University Computer Science Department Stony Brook University [email protected] September 12, 2020 David Gu (Stony Brook University) Computational Conformal Geometry September 12, 2020 1 / 65 Uniformization Figure: Closed surface uniformization. David Gu (Stony Brook University) Computational Conformal Geometry September 12, 2020 2 / 65 Hyperbolic Structure Fundamental Group Suppose (S; g) is a closed high genus surface g > 1. The fundamental group is π1(S; q), represented as −1 −1 −1 −1 π1(S; q) = a1; b1; a2; b2; ; ag ; bg a1b1a b ag bg a b : h ··· j 1 1 ··· g g i Universal Covering Space universal covering space of S is S~, the projection map is p : S~ S.A ! deck transformation is an automorphism of S~, ' : S~ S~, p ' = '. All the deck transformations form the Deck transformation! group◦ DeckS~. ' Deck(S~), choose a pointq ~ S~, andγ ~ S~ connectsq ~ and '(~q). The 2 2 ⊂ projection γ = p(~γ) is a loop on S, then we obtain an isomorphism: Deck(S~) π1(S; q);' [γ] ! 7! David Gu (Stony Brook University) Computational Conformal Geometry September 12, 2020 3 / 65 Hyperbolic Structure Uniformization The uniformization metric is ¯g = e2ug, such that the K¯ 1 everywhere. ≡ − 2 Then (S~; ¯g) can be isometrically embedded on the hyperbolic plane H . The On the hyperbolic plane, all the Deck transformations are isometric transformations, Deck(S~) becomes the so-called Fuchsian group, −1 −1 −1 −1 Fuchs(S) = α1; β1; α2; β2; ; αg ; βg α1β1α β αg βg α β : h ··· j 1 1 ··· g g i The Fuchsian group generators are global conformal invariants, and form the coordinates in Teichm¨ullerspace. -
The Riemann Mapping Theorem Christopher J. Bishop
The Riemann Mapping Theorem Christopher J. Bishop C.J. Bishop, Mathematics Department, SUNY at Stony Brook, Stony Brook, NY 11794-3651 E-mail address: [email protected] 1991 Mathematics Subject Classification. Primary: 30C35, Secondary: 30C85, 30C62 Key words and phrases. numerical conformal mappings, Schwarz-Christoffel formula, hyperbolic 3-manifolds, Sullivan’s theorem, convex hulls, quasiconformal mappings, quasisymmetric mappings, medial axis, CRDT algorithm The author is partially supported by NSF Grant DMS 04-05578. Abstract. These are informal notes based on lectures I am giving in MAT 626 (Topics in Complex Analysis: the Riemann mapping theorem) during Fall 2008 at Stony Brook. We will start with brief introduction to conformal mapping focusing on the Schwarz-Christoffel formula and how to compute the unknown parameters. In later chapters we will fill in some of the details of results and proofs in geometric function theory and survey various numerical methods for computing conformal maps, including a method of my own using ideas from hyperbolic and computational geometry. Contents Chapter 1. Introduction to conformal mapping 1 1. Conformal and holomorphic maps 1 2. M¨obius transformations 16 3. The Schwarz-Christoffel Formula 20 4. Crowding 27 5. Power series of Schwarz-Christoffel maps 29 6. Harmonic measure and Brownian motion 39 7. The quasiconformal distance between polygons 48 8. Schwarz-Christoffel iterations and Davis’s method 56 Chapter 2. The Riemann mapping theorem 67 1. The hyperbolic metric 67 2. Schwarz’s lemma 69 3. The Poisson integral formula 71 4. A proof of Riemann’s theorem 73 5. Koebe’s method 74 6. -
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. -
Elliot Glazer Topological Manifolds
Manifolds and complex structure Elliot Glazer Topological Manifolds • An n-dimensional manifold M is a space that is locally Euclidean, i.e. near any point p of M, we can define points of M by n real coordinates • Curves are 1-dimensional manifolds (1-manifolds) • Surfaces are 2-manifolds • A chart (U, f) is an open subset U of M and a homeomorphism f from U to some open subset of R • An atlas is a set of charts that cover M Some important 2-manifolds A very important 2-manifold Smooth manifolds • We want manifolds to have more than just topological structure • There should be compatibility between intersecting charts • For intersecting charts (U, f), (V, g), we define a natural transition map from one set of coordinates to the other • A manifold is C^k if it has an atlas consisting of charts with C^k transition maps • Of particular importance are smooth maps • Whitney embedding theorem: smooth m-manifolds can be smoothly embedded into R^{2m} Complex manifolds • An n-dimensional complex manifold is a space such that, near any point p, each point can be defined by n complex coordinates, and the transition maps are holomorphic (complex structure) • A manifold with n complex dimensions is also a real manifold with 2n dimensions, but with a complex structure • Complex 1-manifolds are called Riemann surfaces (surfaces because they have 2 real dimensions) • Holoorphi futios are rigid sie they are defied y accumulating sets • Whitney embedding theorem does not extend to complex manifolds Classifying Riemann surfaces • 3 basic Riemann surfaces • The complex plane • The unit disk (distinct from plane by Liouville theorem) • Riemann sphere (aka the extended complex plane), uses two charts • Uniformization theorem: simply connected Riemann surfaces are conformally equivalent to one of these three . -
The Classical Groups and Domains 1. the Disk, Upper Half-Plane, SL 2(R
(June 8, 2018) The Classical Groups and Domains Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ The complex unit disk D = fz 2 C : jzj < 1g has four families of generalizations to bounded open subsets in Cn with groups acting transitively upon them. Such domains, defined more precisely below, are bounded symmetric domains. First, we recall some standard facts about the unit disk, the upper half-plane, the ambient complex projective line, and corresponding groups acting by linear fractional (M¨obius)transformations. Happily, many of the higher- dimensional bounded symmetric domains behave in a manner that is a simple extension of this simplest case. 1. The disk, upper half-plane, SL2(R), and U(1; 1) 2. Classical groups over C and over R 3. The four families of self-adjoint cones 4. The four families of classical domains 5. Harish-Chandra's and Borel's realization of domains 1. The disk, upper half-plane, SL2(R), and U(1; 1) The group a b GL ( ) = f : a; b; c; d 2 ; ad − bc 6= 0g 2 C c d C acts on the extended complex plane C [ 1 by linear fractional transformations a b az + b (z) = c d cz + d with the traditional natural convention about arithmetic with 1. But we can be more precise, in a form helpful for higher-dimensional cases: introduce homogeneous coordinates for the complex projective line P1, by defining P1 to be a set of cosets u 1 = f : not both u; v are 0g= × = 2 − f0g = × P v C C C where C× acts by scalar multiplication. -
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 . -
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. -
A Brief Note on the Approach to the Conic Sections of a Right Circular Cone from Dynamic Geometry
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by EPrints Complutense A brief note on the approach to the conic sections of a right circular cone from dynamic geometry Eugenio Roanes–Lozano Abstract. Nowadays there are different powerful 3D dynamic geometry systems (DGS) such as GeoGebra 5, Calques 3D and Cabri Geometry 3D. An obvious application of this software that has been addressed by several authors is obtaining the conic sections of a right circular cone: the dynamic capabilities of 3D DGS allows to slowly vary the angle of the plane w.r.t. the axis of the cone, thus obtaining the different types of conics. In all the approaches we have found, a cone is firstly constructed and it is cut through variable planes. We propose to perform the construction the other way round: the plane is fixed (in fact it is a very convenient plane: z = 0) and the cone is the moving object. This way the conic is expressed as a function of x and y (instead of as a function of x, y and z). Moreover, if the 3D DGS has algebraic capabilities, it is possible to obtain the implicit equation of the conic. Mathematics Subject Classification (2010). Primary 68U99; Secondary 97G40; Tertiary 68U05. Keywords. Conics, Dynamic Geometry, 3D Geometry, Computer Algebra. 1. Introduction Nowadays there are different powerful 3D dynamic geometry systems (DGS) such as GeoGebra 5 [2], Calques 3D [7, 10] and Cabri Geometry 3D [11]. Focusing on the most widespread one (GeoGebra 5) and conics, there is a “conics” icon in the ToolBar. -
Chapter 9: Conics Section 9.1 Ellipses
Chapter 9: Conics Section 9.1 Ellipses ..................................................................................................... 579 Section 9.2 Hyperbolas ............................................................................................... 597 Section 9.3 Parabolas and Non-Linear Systems ......................................................... 617 Section 9.4 Conics in Polar Coordinates..................................................................... 630 In this chapter, we will explore a set of shapes defined by a common characteristic: they can all be formed by slicing a cone with a plane. These families of curves have a broad range of applications in physics and astronomy, from describing the shape of your car headlight reflectors to describing the orbits of planets and comets. Section 9.1 Ellipses The National Statuary Hall1 in Washington, D.C. is an oval-shaped room called a whispering chamber because the shape makes it possible for sound to reflect from the walls in a special way. Two people standing in specific places are able to hear each other whispering even though they are far apart. To determine where they should stand, we will need to better understand ellipses. An ellipse is a type of conic section, a shape resulting from intersecting a plane with a cone and looking at the curve where they intersect. They were discovered by the Greek mathematician Menaechmus over two millennia ago. The figure below2 shows two types of conic sections. When a plane is perpendicular to the axis of the cone, the shape of the intersection is a circle. A slightly titled plane creates an oval-shaped conic section called an ellipse. 1 Photo by Gary Palmer, Flickr, CC-BY, https://www.flickr.com/photos/gregpalmer/2157517950 2 Pbroks13 (https://commons.wikimedia.org/wiki/File:Conic_sections_with_plane.svg), “Conic sections with plane”, cropped to show only ellipse and circle by L Michaels, CC BY 3.0 This chapter is part of Precalculus: An Investigation of Functions © Lippman & Rasmussen 2020. -
Notes on Euclidean Geometry Kiran Kedlaya Based
Notes on Euclidean Geometry Kiran Kedlaya based on notes for the Math Olympiad Program (MOP) Version 1.0, last revised August 3, 1999 c Kiran S. Kedlaya. This is an unfinished manuscript distributed for personal use only. In particular, any publication of all or part of this manuscript without prior consent of the author is strictly prohibited. Please send all comments and corrections to the author at [email protected]. Thank you! Contents 1 Tricks of the trade 1 1.1 Slicing and dicing . 1 1.2 Angle chasing . 2 1.3 Sign conventions . 3 1.4 Working backward . 6 2 Concurrence and Collinearity 8 2.1 Concurrent lines: Ceva’s theorem . 8 2.2 Collinear points: Menelaos’ theorem . 10 2.3 Concurrent perpendiculars . 12 2.4 Additional problems . 13 3 Transformations 14 3.1 Rigid motions . 14 3.2 Homothety . 16 3.3 Spiral similarity . 17 3.4 Affine transformations . 19 4 Circular reasoning 21 4.1 Power of a point . 21 4.2 Radical axis . 22 4.3 The Pascal-Brianchon theorems . 24 4.4 Simson line . 25 4.5 Circle of Apollonius . 26 4.6 Additional problems . 27 5 Triangle trivia 28 5.1 Centroid . 28 5.2 Incenter and excenters . 28 5.3 Circumcenter and orthocenter . 30 i 5.4 Gergonne and Nagel points . 32 5.5 Isogonal conjugates . 32 5.6 Brocard points . 33 5.7 Miscellaneous . 34 6 Quadrilaterals 36 6.1 General quadrilaterals . 36 6.2 Cyclic quadrilaterals . 36 6.3 Circumscribed quadrilaterals . 38 6.4 Complete quadrilaterals . 39 7 Inversive Geometry 40 7.1 Inversion . -
Math 372: Fall 2015: Solutions to Homework
Math 372: Fall 2015: Solutions to Homework Steven Miller December 7, 2015 Abstract Below are detailed solutions to the homeworkproblemsfrom Math 372 Complex Analysis (Williams College, Fall 2015, Professor Steven J. Miller, [email protected]). The course homepage is http://www.williams.edu/Mathematics/sjmiller/public_html/372Fa15 and the textbook is Complex Analysis by Stein and Shakarchi (ISBN13: 978-0-691-11385-2). Note to students: it’s nice to include the statement of the problems, but I leave that up to you. I am only skimming the solutions. I will occasionally add some comments or mention alternate solutions. If you find an error in these notes, let me know for extra credit. Contents 1 Math 372: Homework #1: Yuzhong (Jeff) Meng and Liyang Zhang (2010) 3 1.1 Problems for HW#1: Due September 21, 2015 . ................ 3 1.2 SolutionsforHW#1: ............................... ........... 3 2 Math 372: Homework #2: Solutions by Nick Arnosti and Thomas Crawford (2010) 8 3 Math 372: Homework #3: Carlos Dominguez, Carson Eisenach, David Gold 12 4 Math 372: Homework #4: Due Friday, October 12, 2015: Pham, Jensen, Kolo˘glu 16 4.1 Chapter3,Exercise1 .............................. ............ 16 4.2 Chapter3,Exercise2 .............................. ............ 17 4.3 Chapter3,Exercise5 .............................. ............ 19 4.4 Chapter3Exercise15d ..... ..... ...... ..... ...... .. ............ 22 4.5 Chapter3Exercise17a ..... ..... ...... ..... ...... .. ............ 22 4.6 AdditionalProblem1 .............................. ............ 22 5 Math 372: Homework #5: Due Monday October 26: Pegado, Vu 24 6 Math 372: Homework #6: Kung, Lin, Waters 34 7 Math 372: Homework #7: Due Monday, November 9: Thompson, Schrock, Tosteson 42 7.1 Problems. ....................................... ......... 46 1 8 Math 372: Homework #8: Thompson, Schrock, Tosteson 47 8.1 Problems.