C Is Proper. Clearly, C ∩ C Is Also the Intersection of C with Any
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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. -
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. -
Section 6.2 Introduction to Conics: Parabolas 103
Section 6.2 Introduction to Conics: Parabolas 103 Course Number Section 6.2 Introduction to Conics: Parabolas Instructor Objective: In this lesson you learned how to write the standard form of the equation of a parabola. Date Important Vocabulary Define each term or concept. Directrix A fixed line in the plane from which each point on a parabola is the same distance as the distance from the point to a fixed point in the plane. Focus A fixed point in the plane from which each point on a parabola is the same distance as the distance from the point to a fixed line in the plane. Focal chord A line segment that passes through the focus of a parabola and has endpoints on the parabola. Latus rectum The specific focal chord perpendicular to the axis of a parabola. Tangent A line is tangent to a parabola at a point on the parabola if the line intersects, but does not cross, the parabola at the point. I. Conics (Page 433) What you should learn How to recognize a conic A conic section, or conic, is . the intersection of a plane as the intersection of a and a double-napped cone. plane and a double- napped cone Name the four basic conic sections: circle, ellipse, parabola, and hyperbola. In the formation of the four basic conics, the intersecting plane does not pass through the vertex of the cone. When the plane does pass through the vertex, the resulting figure is a(n) degenerate conic , such as . a point, a line, or a pair of intersecting lines. -
Projective Geometry Lecture Notes
Projective Geometry Lecture Notes Thomas Baird March 26, 2014 Contents 1 Introduction 2 2 Vector Spaces and Projective Spaces 4 2.1 Vector spaces and their duals . 4 2.1.1 Fields . 4 2.1.2 Vector spaces and subspaces . 5 2.1.3 Matrices . 7 2.1.4 Dual vector spaces . 7 2.2 Projective spaces and homogeneous coordinates . 8 2.2.1 Visualizing projective space . 8 2.2.2 Homogeneous coordinates . 13 2.3 Linear subspaces . 13 2.3.1 Two points determine a line . 14 2.3.2 Two planar lines intersect at a point . 14 2.4 Projective transformations and the Erlangen Program . 15 2.4.1 Erlangen Program . 16 2.4.2 Projective versus linear . 17 2.4.3 Examples of projective transformations . 18 2.4.4 Direct sums . 19 2.4.5 General position . 20 2.4.6 The Cross-Ratio . 22 2.5 Classical Theorems . 23 2.5.1 Desargues' Theorem . 23 2.5.2 Pappus' Theorem . 24 2.6 Duality . 26 3 Quadrics and Conics 28 3.1 Affine algebraic sets . 28 3.2 Projective algebraic sets . 30 3.3 Bilinear and quadratic forms . 31 3.3.1 Quadratic forms . 33 3.3.2 Change of basis . 33 1 3.3.3 Digression on the Hessian . 36 3.4 Quadrics and conics . 37 3.5 Parametrization of the conic . 40 3.5.1 Rational parametrization of the circle . 42 3.6 Polars . 44 3.7 Linear subspaces of quadrics and ruled surfaces . 46 3.8 Pencils of quadrics and degeneration . 47 4 Exterior Algebras 52 4.1 Multilinear algebra . -
Stable Rationality of Quadric Surface Bundles Over Surfaces
STABLE RATIONALITY OF QUADRIC SURFACE BUNDLES OVER SURFACES BRENDAN HASSETT, ALENA PIRUTKA, AND YURI TSCHINKEL Abstract. We study rationality properties of quadric surface bun- dles over the projective plane. We exhibit families of smooth pro- jective complex fourfolds of this type over connected bases, con- taining both rational and non-rational fibers. 1. Introduction Is a deformation of a smooth rational (or irrational) projective vari- ety still rational (or irrational)? The main goal of this paper is to show that rationality is not deformation invariant for families of smooth com- plex projective varieties of dimension four. Examples along these lines are known when singular fibers are allowed, e.g., smooth cubic three- folds (which are irrational) may specialize to cubic threefolds with or- dinary double points (which are rational), while smooth cubic surfaces (which are rational) may specialize to cones over elliptic curves. Totaro shows that specializations of rational varieties need not be rational in higher dimensions if mild singularities are allowed [Tot16b]. However, de Fernex and Fusi [dFF13] show that the locus of rational fibers in a smooth family of projective complex threefolds is a countable union of closed subsets on the base. Let S be a smooth projective rational surface over the complex num- bers with function field K = C(S). A quadric surface bundle consists of a fourfold X and a flat projective morphism π : X ! S such that the generic fiber Q=K of π is a smooth quadric surface. We assume that π factors through the projectivization of a rank four vector bundle on S such that the fibers are (possibly singular) quadric surfaces; see Section 3 for relevant background. -
Single View Geometry Camera Model & Orientation + Position Estimation
Single View Geometry Camera model & Orientation + Position estimation What am I? Vanishing point Mapping from 3D to 2D Point & Line Goal: Homogeneous coordinates Point – represent coordinates in 2 dimensions with a 3-vector &x# &x# homogeneous coords $y! $ ! ''''''→$ ! %y" %$1"! The projective plane • Why do we need homogeneous coordinates? – represent points at infinity, homographies, perspective projection, multi-view relationships • What is the geometric intuition? – a point in the image is a ray in projective space y (sx,sy,s) (x,y,1) (0,0,0) z x image plane • Each point (x,y) on the plane is represented by a ray (sx,sy,s) – all points on the ray are equivalent: (x, y, 1) ≅ (sx, sy, s) Projective Lines Projective lines • What does a line in the image correspond to in projective space? • A line is a plane of rays through origin – all rays (x,y,z) satisfying: ax + by + cz = 0 ⎡x⎤ in vector notation : 0 a b c ⎢y⎥ = [ ]⎢ ⎥ ⎣⎢z⎦⎥ l p • A line is also represented as a homogeneous 3-vector l Line Representation • a line is • is the distance from the origin to the line • is the norm direction of the line • It can also be written as Example of Line Example of Line (2) 0.42*pi Homogeneous representation Line in Is represented by a point in : But correspondence of line to point is not unique We define set of equivalence class of vectors in R^3 - (0,0,0) As projective space Projective lines from two points Line passing through two points Two points: x Define a line l is the line passing two points Proof: Line passing through two points • More -
Read to the Infinite and Back Again
TO THE INFINITE AND BACK AGAIN Part I Henrike Holdrege A Workbook in Projective Geometry To the Infinite and Back Again A Workbook in Projective Geometry Part I HENRIKE HOLDREGE EVOLVING SCIENCE ASSOCIATION ————————————————— A collaboration of The Nature Institute and The Myrin Institute 2019 Copyright 2019 Henrike Holdrege All rights reserved. Second printing, 2019 Published by the Evolving Science Association, a collaboration of The Nature Institute and The Myrin Institute The Nature Institute: 20 May Hill Road, Ghent, NY 12075 natureinstitute.org The Myrin Institute: 187 Main Street, Great Barrington, MA 01230 myrin.org ISBN 978-0-9744906-5-6 Cover painting by Angelo Andes, Desargues 10 Lines 10 Points This book, and other Nature Institute publications, can be purchased from The Nature Institute’s online bookstore: natureinstitute.org/store or contact The Nature Institute: [email protected] or 518-672-0116. Table of Contents INTRODUCTION 1 PREPARATIONS 4 TEN BASIC ENTITIES 6 PRELUDE Form and Forming 7 CHAPTER 1 The Harmonic Net and the Harmonic Four Points 11 INTERLUDE The Infinitely Distant Point of a Line 23 CHAPTER 2 The Theorem of Pappus 27 INTERLUDE A Triangle Transformation 36 CHAPTER 3 Sections of the Point Field 38 INTERLUDE The Projective versus the Euclidean Point Field 47 CHAPTER 4 The Theorem of Desargues 49 INTERLUDE The Line at Infinity 61 CHAPTER 5 Desargues’ Theorem in Three-dimensional Space 64 CHAPTER 6 Shadows, Projections, and Linear Perspective 78 CHAPTER 7 Homologies 89 INTERLUDE The Plane at Infinity 96 CLOSING 99 ACKNOWLEDGMENTS 101 BIBLIOGRAPHY 103 I n t r o d u c t i o n My intent in this book is to allow you, the reader, to engage with the fundamental concepts of projective geometry. -
Where Parallel Lines Meet
Where Parallel Lines WhereMeet parallel lines meet ...a geometric love story Renzo Cavalieri Renzo Cavalieri Colorado State University MATH DAY 2009 University of Northern Colorado Oct 22, 2014 Renzo Cavalieri Where parallel lines meet... Renzo Cavalieri Where Parallel Lines Meet Geometry γ"!µετρια \Geos = earth" + \Metron = measure" Was developed by King Sesostris as a way to counter tax fraud! Renzo Cavalieri Where Parallel Lines Meet Abstraction Abstraction In order to study in broader generalityIn order to the study properties in broader of spaces, __________ mathematiciansgenerality the properties created concepts of spaces, thatmathematicians abstract and created generalize concepts our physicalthat abstract experience. and generalize our Forphysical example, experience. a line or a plane do not reallyFor example, exist in our a line world!or a plane do not really exist in our world! Renzo Cavalieri Where Parallel Lines Meet Renzo Cavalieri Where Parallel Lines Meet Euclid's Posulates 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 Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4 All right angles are congruent. 5 Given a line m and a point P not belonging to m, there is a unique line through P that never intersects m. Renzo Cavalieri Where Parallel Lines Meet The parallel postulate The parallel postulate was first shaken by the following puzzle: A hunter gets off her pickup and walks south for one mile. Then she makes a 90◦ left turn and walks east for another mile. -
Euclidean Versus Projective Geometry
Projective Geometry Projective Geometry Euclidean versus Projective Geometry n Euclidean geometry describes shapes “as they are” – Properties of objects that are unchanged by rigid motions » Lengths » Angles » Parallelism n Projective geometry describes objects “as they appear” – Lengths, angles, parallelism become “distorted” when we look at objects – Mathematical model for how images of the 3D world are formed. Projective Geometry Overview n Tools of algebraic geometry n Informal description of projective geometry in a plane n Descriptions of lines and points n Points at infinity and line at infinity n Projective transformations, projectivity matrix n Example of application n Special projectivities: affine transforms, similarities, Euclidean transforms n Cross-ratio invariance for points, lines, planes Projective Geometry Tools of Algebraic Geometry 1 n Plane passing through origin and perpendicular to vector n = (a,b,c) is locus of points x = ( x 1 , x 2 , x 3 ) such that n · x = 0 => a x1 + b x2 + c x3 = 0 n Plane through origin is completely defined by (a,b,c) x3 x = (x1, x2 , x3 ) x2 O x1 n = (a,b,c) Projective Geometry Tools of Algebraic Geometry 2 n A vector parallel to intersection of 2 planes ( a , b , c ) and (a',b',c') is obtained by cross-product (a'',b'',c'') = (a,b,c)´(a',b',c') (a'',b'',c'') O (a,b,c) (a',b',c') Projective Geometry Tools of Algebraic Geometry 3 n Plane passing through two points x and x’ is defined by (a,b,c) = x´ x' x = (x1, x2 , x3 ) x'= (x1 ', x2 ', x3 ') O (a,b,c) Projective Geometry Projective Geometry -
A Gallery of Conics by Five Elements
Forum Geometricorum Volume 14 (2014) 295–348. FORUM GEOM ISSN 1534-1178 A Gallery of Conics by Five Elements Paris Pamfilos Abstract. This paper is a review on conics defined by five elements, i.e., either lines to which the conic is tangent or points through which the conic passes. The gallery contains all cases combining a number (n) of points and a number (5 − n) of lines and also allowing coincidences of some points with some lines. Points and/or lines at infinity are handled as particular cases. 1. Introduction In the following we review the construction of conics by five elements: points and lines, briefly denoted by (αP βT ), with α + β =5. In these it is required to construct a conic passing through α given points and tangent to β given lines. The six constructions, resulting by giving α, β the values 0 to 5 and considering the data to be in general position, are considered the most important ([18, p. 387]), and are to be found almost on every book about conics. It seems that constructions for which some coincidences are allowed have attracted less attention, though they are related to many interesting theorems of the geometry of conics. Adding to the six main cases those with the projectively different possible coincidences we land to 12 main constructions figuring on the first column of the classifying table below. The six added cases can be considered as limiting cases of the others, in which a point tends to coincide with another or with a line. The twelve main cases are the projectively inequivalent to which every other case can be reduced by means of a projectivity of the plane. -
Pencils of Conics
Chaos\ Solitons + Fractals Vol[ 8\ No[ 6\ pp[ 0960Ð0975\ 0887 Þ 0887 Elsevier Science Ltd[ All rights reserved Pergamon Printed in Great Britain \ 9859Ð9668:87 ,08[99 ¦ 9[99 PII] S9859!9668"86#99978!0 Pencils of Conics] a Means Towards a Deeper Understanding of the Arrow of Time< METOD SANIGA Astronomical Institute\ Slovak Academy of Sciences\ SK!948 59 Tatranska Lomnica\ The Slovak Republic "Accepted 7 April 0886# Abstract*The paper aims at giving a su.ciently complex description of the theory of pencil!generated temporal dimensions in a projective plane over reals[ The exposition starts with a succinct outline of the mathematical formalism and goes on with introducing the de_nitions of pencil!time and pencil!space\ both at the abstract "projective# and concrete "a.ne# levels[ The structural properties of all possible types of temporal arrows are analyzed and\ based on symmetry principles\ the uniqueness of that mimicking best the reality is justi_ed[ A profound connection between the character of di}erent {ordinary| arrows and the number of spatial dimensions is revealed[ Þ 0887 Elsevier Science Ltd[ All rights reserved 0[ INTRODUCTION The depth of our penetration into the problem is measured not by the state of things on the frontier\ but rather how far away the frontier lies[ One of the most pronounced examples of such a situation concerns\ beyond any doubt\ the nature of time[ What we have precisely in mind here is a {{puzzling discrepancy between our perception of time and what modern physical theory tells us to believe|| ð0Ł\ see -
THE HESSE PENCIL of PLANE CUBIC CURVES 1. Introduction In
THE HESSE PENCIL OF PLANE CUBIC CURVES MICHELA ARTEBANI AND IGOR DOLGACHEV Abstract. This is a survey of the classical geometry of the Hesse configuration of 12 lines in the projective plane related to inflection points of a plane cubic curve. We also study two K3surfaceswithPicardnumber20whicharise naturally in connection with the configuration. 1. Introduction In this paper we discuss some old and new results about the widely known Hesse configuration of 9 points and 12 lines in the projective plane P2(k): each point lies on 4 lines and each line contains 3 points, giving an abstract configuration (123, 94). Through most of the paper we will assume that k is the field of complex numbers C although the configuration can be defined over any field containing three cubic roots of unity. The Hesse configuration can be realized by the 9 inflection points of a nonsingular projective plane curve of degree 3. This discovery is attributed to Colin Maclaurin (1698-1746) (see [46], p. 384), however the configuration is named after Otto Hesse who was the first to study its properties in [24], [25]1. In particular, he proved that the nine inflection points of a plane cubic curve form one orbit with respect to the projective group of the plane and can be taken as common inflection points of a pencil of cubic curves generated by the curve and its Hessian curve. In appropriate projective coordinates the Hesse pencil is given by the equation (1) λ(x3 + y3 + z3)+µxyz =0. The pencil was classically known as the syzygetic pencil 2 of cubic curves (see [9], p.