Notes on Multi-View Geometry in Computer Vision
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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. -
Lecture 3: Geometry
E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010-2015 Lecture 3: Geometry Geometry is the science of shape, size and symmetry. While arithmetic dealt with numerical structures, geometry deals with metric structures. Geometry is one of the oldest mathemati- cal disciplines and early geometry has relations with arithmetics: we have seen that that the implementation of a commutative multiplication on the natural numbers is rooted from an inter- pretation of n × m as an area of a shape that is invariant under rotational symmetry. Number systems built upon the natural numbers inherit this. Identities like the Pythagorean triples 32 +42 = 52 were interpreted geometrically. The right angle is the most "symmetric" angle apart from 0. Symmetry manifests itself in quantities which are invariant. Invariants are one the most central aspects of geometry. Felix Klein's Erlanger program uses symmetry to classify geome- tries depending on how large the symmetries of the shapes are. In this lecture, we look at a few results which can all be stated in terms of invariants. In the presentation as well as the worksheet part of this lecture, we will work us through smaller miracles like special points in triangles as well as a couple of gems: Pythagoras, Thales,Hippocrates, Feuerbach, Pappus, Morley, Butterfly which illustrate the importance of symmetry. Much of geometry is based on our ability to measure length, the distance between two points. A modern way to measure distance is to determine how long light needs to get from one point to the other. This geodesic distance generalizes to curved spaces like the sphere and is also a practical way to measure distances, for example with lasers. -
Notes 20: Afine Geometry
Notes 20: Afine Geometry Example 1. Let C be the curve in R2 defined by x2 + 4xy + y2 + y2 − 1 = 0 = (x + 2y)2 + y2 − 1: If we set u = x + 2y; v = y; we obtain u2 + v2 − 1 = 0 which is a circle in the u; v plane. Let 2 2 F : R ! R ; (x; y) 7! (u = 2x + y; v = y): We look at the geometry of F: Angles: This transformation does not preserve angles. For example the line 2x + y = 0 is mapped to the line u = 0; and the line y = 0 is mapped to v = 0: The two lines in the x − y plane are not orthogonal, while the lines in the u − v plane are orthogonal. Distances: This transformation does not preserve distance. The point A = (−1=2; 1) is mapped to a = (0; 1) and the point B = (0; 0) is mapped to b = (0; 0): the distance AB is not equal to the distance ab: Circles: The curve C is an ellipse with axis along the lines 2x + y = 0 and x = 0: It is mapped to the unit circle in the u − v plane. Lines: The map is a one-to-one onto map that maps lines to lines. What is a Geometry? We can think of a geometry as having three parts: A set of points, a special set of subsets called lines, and a group of transformations of the set of points that maps lines to lines. In the case of Euclidean geometry, the set of points is the familiar plane R2: The special subsets are what we ordinarily call lines. -
(Finite Affine Geometry) References • Bennett, Affine An
MATHEMATICS 152, FALL 2003 METHODS OF DISCRETE MATHEMATICS Outline #7 (Finite Affine Geometry) References • Bennett, Affine and Projective Geometry, Chapter 3. This book, avail- able in Cabot Library, covers all the proofs and has nice diagrams. • “Faculty Senate Affine Geometry”(attached). This has all the steps for each proof, but no diagrams. There are numerous references to diagrams on the course Web site, however, and the combination of this document and the Web site should be all that you need. • The course Web site, AffineDiagrams folder. This has links that bring up step-by-step diagrams for all key results. These diagrams will be available in class, and you are welcome to use them instead of drawing new diagrams on the blackboard. • The Windows application program affine.exe, which can be downloaded from the course Web site. • Data files for the small, medium, and large affine senates. These accom- pany affine.exe, since they are data files read by that program. The file affine.zip has everything. • PHP version of the affine geometry software. This program, written by Harvard undergraduate Luke Gustafson, runs directly off the Web and generates nice diagrams whenever you use affine geometry to do arithmetic. It also has nice built-in documentation. Choose the PHP- Programs folder on the Web site. 1. State the first four of the five axioms for a finite affine plane, using the terms “instructor” and “committee” instead of “point” and “line.” For A4 (Desargues), draw diagrams (or show the ones on the Web site) to illustrate the two cases (three parallel lines and three concurrent lines) in the Euclidean plane. -
Lecture 5: Affine Graphics a Connect the Dots Approach to Two-Dimensional Computer Graphics
Lecture 5: Affine Graphics A Connect the Dots Approach to Two-Dimensional Computer Graphics The lines are fallen unto me in pleasant places; Psalms 16:6 1. Two Shortcomings of Turtle Graphics Two points determine a line. In Turtle Graphics we use this simple fact to draw a line joining the two points at which the turtle is located before and after the execution of each FORWARD command. By programming the turtle to move about and to draw lines in this fashion, we are able to generate some remarkable figures in the plane. Nevertheless, the turtle has two annoying idiosyncrasies. First, the turtle has no memory, so the order in which the turtle encounters points is crucial. Thus, even though the turtle leaves behind a trace of her path, there is no direct command in LOGO to return the turtle to an arbitrary previously encountered location. Second, the turtle is blissfully unaware of the outside universe. The turtle carries her own local coordinate system -- her state -- but the turtle does not know her position relative to any other point in the plane. Turtle geometry is a local, intrinsic geometry; the turtle knows nothing of the extrinsic, global geometry of the external world. The turtle can draw a circle, but the turtle has no idea where the center of the circle might be or even that there is such a concept as a center, a point outside her path around the circumference. These two shortcomings -- no memory and no knowledge of the outside world -- often make the turtle cumbersome to program. -
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 -
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 . -
Affine Geometry
CHAPTER II AFFINE GEOMETRY In the previous chapter we indicated how several basic ideas from geometry have natural interpretations in terms of vector spaces and linear algebra. This chapter continues the process of formulating basic geometric concepts in such terms. It begins with standard material, moves on to consider topics not covered in most courses on classical deductive geometry or analytic geometry, and it concludes by giving an abstract formulation of the concept of geometrical incidence and closely related issues. 1. Synthetic affine geometry In this section we shall consider some properties of Euclidean spaces which only depend upon the axioms of incidence and parallelism Definition. A three-dimensional incidence space is a triple (S; L; P) consisting of a nonempty set S (whose elements are called points) and two nonempty disjoint families of proper subsets of S denoted by L (lines) and P (planes) respectively, which satisfy the following conditions: (I { 1) Every line (element of L) contains at least two points, and every plane (element of P) contains at least three points. (I { 2) If x and y are distinct points of S, then there is a unique line L such that x; y 2 L. Notation. The line given by (I {2) is called xy. (I { 3) If x, y and z are distinct points of S and z 62 xy, then there is a unique plane P such that x; y; z 2 P . (I { 4) If a plane P contains the distinct points x and y, then it also contains the line xy. (I { 5) If P and Q are planes with a nonempty intersection, then P \ Q contains at least two points. -
References Finite Affine Geometry
MATHEMATICS S-152, SUMMER 2005 THE MATHEMATICS OF SYMMETRY Outline #7 (Finite Affine Geometry) References • Bennett, Affine and Projective Geometry, Chapter 3. This book, avail- able in Cabot Library, covers all the proofs and has nice diagrams. • “Faculty Senate Affine Geometry” (attached). This has all the steps for each proof, but no diagrams. There are numerous references to diagrams on the course web site; however, and the combination of this document and the web site should be all that you need. • The course web site, AffineDiagrams folder. This has links that bring up step-by-step diagrams for all key results. These diagrams will be available in class, and you are welcome to use them instead of drawing new diagrams on the blackboard. • The Windows application program, affine.exe, which can be down- loaded from the course web site. • Data files for the small, medium, and large affine senates. These ac- company affine.exe, since they are data files read by that program. The file affine.zip has everything. • PHP version of the affine geometry software. This program, written by Harvard undergraduate Luke Gustafson, runs directly off the web and generates nice diagrams whenever you use affine geometry to do arithmetic. It also has nice built-in documentation. Choose the PHP- Programs folder on the web site. Finite Affine Geometry 1. State the first four of the five axioms for a finite affine plane, using the terms “instructor” and “committee” instead of “point” and “line.” For A4 (Desargues), draw diagrams (or show the ones on the web site) to 1 illustrate the two cases (three parallel lines and three concurrent lines) in the Euclidean plane. -
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 -
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.