Projective Geometry: A Short Introduction

1 Projective geometry : A Short IntroductionLecture NotesEdmond BoyerMaster MOSIGI ntroduction to Projective GeometryContents1 Objective .. Historical Background .. Bibliography ..42 Projective Definitions .. Properties .. The hyperplane at infinity .. 123 The Projective Introduction .. Projective transformation ofP1.. The cross-ratio .. 144 The Projective Points and lines .. Line at infinity .. Homographies .. Conics .. Affine transformations .. Euclidean transformations .. Particular transformations .. Transformation hierarchy .. 25 Grenoble Universities1 Master MOSIGI ntroduction to Projective GeometryChapter ObjectiveThe objective of this course is to give basic notions and intuitions onprojectivegeometry. The interest of Projective geometry arises in several visual comput-ing domains, in particular computer vision modelling and computer provides a mathematical formalism to describe the geometry of cameras andthe associated transformations, hence enabling the design of computational ap-proaches that manipulates 2D projections of 3D objects.

2 In that respect, afundamental aspect is the fact that objects at infinity can be represented andmanipulated with Projective geometry and this in contrast to the Euclideangeometry. This allows perspective deformations to be represented as : Example of perspective deformation or 2D Projective argument is that Euclidean geometry is sometimes difficult to use inalgorithms, with particular cases arising from non-generic situations ( twoparallel lines never intersect) that must be identified. In contrast, projectivegeometry generalizes several definitions and properties, two lines alwaysintersect (see fig. ). It allows also to represent any transformation that pre-serves coincidence relationships in a matrix form ( perspective projections)that is easier to use, in particular in computer Universities2 Master MOSIGI ntroduction to Projective GeometryInfinitynon-parallel linesparallel linesFigure : Line intersections in a Projective Historical BackgroundThe origins of geometry date back to Egypt and Babylon (2000 BC).

3 It wasfirst designed to address problems of everyday life, such as area estimations andconstruction, but abstract notions were missing. around 600 BC: The familiar form of geometry begins in Greece. Firstabstract notions appear, especially the notion of infinite space. 300 BC:Euclide, in the bookElements, introduces an axiomatic ap-proach to geometry . From axioms, grounded on evidences or the experi-ence, one can infer theorems. The Euclidean geometry is based on mea-sures taken on rigid shapes, lengths and angles, hence the notion ofshape invariance (under rigid motion) and also that (Euclidean) geometricproperties are invariant under rigid motions. 15th century: the Euclidean geometry is not sufficient to model perspec-tive deformations. Painters and architects start manipulating the notionof perspective. An open question then is what are the properties sharedby two perspective views of the same scene ?

4 17th century:Desargues (architect and engineer) describes conics as per-spective deformations of the circle. He considers the point at infinity asthe intersection of parallel lines. 18th century:Descartes, Fermatcontrast the synthetic geometry of theGreeks, based on primitives with the analytical geometry , based instead oncoordinates. Desargue s ideas are taken up byPascal, among others, whohowever focuses on infinitesimal approaches and Cartesian the descriptive geometry and study in particular theconservation of angles and lengths in projections. 19th century:Poncelet(a Napoleon officer) writes, in 1822, a treaty onprojective properties of figures and the invariance by projection. This isthe first treaty on Projective geometry : a Projective property is a prop-erty invariant by et M obiusstudy the most generalGrenoble Universities3 Master MOSIGI ntroduction to Projective Geometryprojective transformations that transform points into points and lines intolines and preserve the cross ratio (the collineations).

5 In 1872,Felix Kleinproposes the Erlangen program, at the Erlangen university, within whicha geometry is not defined by the objects it represents but by their trans-formations, hence the study of invariants for a group of yields a hierarchy of geometries, defined as groups of transformations,where the Euclidean geometry is part of the affine geometry which is itselfincluded into the Projective GeometryAffine GeometryEuclidean GeometryFigure : The geometry BibliographyThe books below served as references for these notes. They include computervision books that present comprehensive chapters on Projective geometry . Semple and Kneebone,Algebraic Projective geometry , ClarendonPress, Oxford (1952) R. Hartley and A. Zisserman,Multiple View geometry , Cambridge Uni-versity Press (2000) O. Faugeras and Q-T. Luong,The geometry of Multiple Images, MITP ress (2001) D.

6 Forsyth and J. Ponce,Computer Vision: A Modern Approach, PrenticeHall (2003)Grenoble Universities4 Master MOSIGI ntroduction to Projective GeometryChapter 2 Projective SpacesIn this chapter, formal definitions and properties of Projective spaces are given,regardless of the dimension. Specific cases such as the line and the plane arestudied in subsequent DefinitionsConsider the real vector spaceRn+1of dimensionn+ 1. Letvbe a non-zeroelement ofRn+1then the setRvof all vectorskv,k R is called a ray ( ).kvFigure : The rayRvis the set of all non-zero vectorskvwith (Geometric Definition)The real Projective spacePn, of dimen-sionn, associated toRn+1is the set of rays ofRn+1. An element ofPnis calleda point and a set of linearly independent (respectively dependent) points ofPnis defined by a set of linearly independent (respectively dependent) Universities5 Master MOSIGI ntroduction to Projective GeometryABCABCRRRF igure : The Projective space associated toR3is called the Projective (Algebraic Definition)A point of a real Projective spacePnisrepresented by a vector of real coordinatesX= [x0.]

7 ,xn]t, at least one ofwhich is non-zero. The{xi}s are called the Projective or homogeneous coordi-nates and two vectorsXandYrepresent the same point when there exists ascalark R such that:xi=kyi i,which we will denote by:X the Projective coordinates of a point are defined up to a scale factorand the correspondence between points and coordinate vectors is not coordinates relate to a Projective basis:Definition ( Projective Basis)A Projective basis is a set of(n+ 2)pointsofPn, no(n+ 1)of which are linearly dependent. For example: .. AnA Grenoble Universities6 Master MOSIGI ntroduction to Projective Geometryis the canonical basis where the{Ai}s are called the basis points andA the relationship between Projective coordinates and a Projective basis is CoordinatesLet{A0,..,An,A }be a basis ofPnwith associatedraysRviandRv respectively.

8 Then for any pointAofPnwith an associatedrayRv, its Projective coordinates [x0,..,xn]tare such that:v=x0v0+..+xnvn,where the scales of the vectors{vi}s associated to the{Ai}s are given by:v =v0+..+vn,which determines the{xi}s up to a scale on Projective coordinatesTo better understand the above characterization of the Projective coor-dinates, let us consider any (n+ 1) vectorsviassociated to the{Ai} definition they form a basis ofRn+1and any vectorvin this spacecan be uniquely decomposed as:v=u0v0+..+unvn, ui R determined by a single set of coordinates{ui}in the vectorbasis{vi}. However the above unique decomposition with theui s doesnot transfer to the associated pointsAand{Ai}s ofPnsince the corre-spondence between points inPnand vectors inRn+1is not instance replacing in the decompositionu0byu0/2 andv0by 2v0still relatesAwith the{Ai}s but with a different set of{ui}s.

9 In orderto uniquely determine the decomposition, let us consider the additionalpointA and letv be one of its associated vector inRn+1then:v =u 0v0+..+u nvn=v0+..+ abovescaledvectorsviare well defined as soon asui6= 0, i(trueby the fact that, by definition,A is linearly independent of any subsetofnpointsAi). Then, any vectorvassociated toAwrites:v=x0v0+..+xnvn,where the{xi}s can vary only by a global scale factor function of thescales ofvandv .Grenoble Universities7 Master MOSIGI ntroduction to Projective GeometryDefinition ( Projective Transformations)A matrixMof dimensions(n+1) (n+ 1)such thatdet(M)6= 0, or equivalently non-singular, defines a lineartransformation fromPnto itself that is called a homography, a collineation ora Projective transformations are the most general transformations that pre-serve incidence relationships, collinearity and PropertiesSome classical and fundamental properties of Projective spaces ofPnthat are linearly independent withm < n.

10 The set of points inPnthat are linearly dependent on thesempointsform a Projective space of dimensionm 1. When this dimension is equal to1,2andn 1, this space is called line, plane and hyperplane respectively. The setof subspaces ofPnwith the same dimension is also a Projective are hyperplanes ofP2and they form a Projective space ofdimension (Duality)The set of hyperplanes of a Projective spacePnis aprojective space of dimensionn. Any definition, property or theorem that appliesto the points of a Projective space is also valid for its lines define a point2 points define a lineFigure : Lines and points are dual and lines are dual in the Projective plane, 2points definea lineis dual to 2lines define a point. Another interesting illustration of theduality is the Desargues theorem (see figure ) that writes:If2triangles are such that the lines joining their corresponding verticesare concurrent then the points of intersections of the corresponding edges areGrenoble Universities8 Master MOSIGI ntroduction to Projective Geometrycollinear,and which reciprocal is its dual (replace in the statementlines joiningwithpoints of intersections of,verticeswithedgesandconcurrentwithcol linearandvice versa).