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Projective Geometry: A Short Introduction

Lecture Notes

Edmond BoyerMaster 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 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.3 The hyperplane at innity . . . . . . . . . . . . . . . . . . . . . .

12

3 The projective line 13

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

3.2 Projective transformation ofP1. . . . . . . . . . . . . . . . . . .14

3.3 The cross-ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

4 The projective plane 17

4.1 Points and lines . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

4.2 Line at innity . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

4.3 Homographies . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

4.4 Conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

4.5 Ane transformations . . . . . . . . . . . . . . . . . . . . . . . .

22

4.6 Euclidean transformations . . . . . . . . . . . . . . . . . . . . . .

22

4.7 Particular transformations . . . . . . . . . . . . . . . . . . . . . .

24

4.8 Transformation hierarchy . . . . . . . . . . . . . . . . . . . . . .

25
Grenoble Universities 1Master MOSIG Introduction to Projective Geometry

Chapter 1

Introduction

1.1 Objective

The objective of this course is to give basic notions and intuitions onprojective 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 innity 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 dicult to use in algorithms, with particular cases arising from non-generic situations (e.g. two parallel lines never intersect) that must be identied. In contrast, projective geometry generalizes several denitions and properties, e.g. two lines always intersect (see g. 1.2). It allows also to represent any transformation that pre- serves coincidence relationships in a matrix form (e.g. perspective projections) that is easier to use, in particular in computer programs. Grenoble Universities 2Master MOSIG Introduction to Projective Geometry Figure 1.2: Line intersections in a projective space

1.2 Historical Background

The origins of geometry date back to Egypt and Babylon (2000 BC). It was rst designed to address problems of everyday life, such as area estimations and construction, but abstract notions were missing. around 600 BC: The familiar form of geometry begins in Greece. First abstract notions appear, especially the notion of innite 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, e.g. lengths and angles, hence the notion of shape invariance (under rigid motion) and also that (Euclidean) geometric properties are invariant under rigid motions.

15th century: the Euclidean geometry is not sucient to model perspec-

tive deformations. Painters and architects start manipulating the notion of perspective. An open question then is "what are the properties shared by two perspective views of the same scene ?"

17th century:Desargues (architect and engineer) describes conics as per-

spective deformations of the circle. He considers the point at innity as the intersection of parallel lines.

18th century:Descartes, Fermatcontrast the synthetic geometry of the

Greeks, based on primitives with the analytical geometry, based instead on coordinates. Desargue's ideas are taken up byPascal, among others, who however focuses on innitesimal approaches and Cartesian coordinates. Mongeintroduces the descriptive geometry and study in particular the conservation of angles and lengths in projections.

19th century:Poncelet(a Napoleon ocer) writes, in 1822, a treaty on

projective properties of gures and the invariance by projection. This is the rst treaty on projective geometry: a projective property is a prop- erty invariant by projection.Chasles et Mobiusstudy the most general

Grenoble Universities 3Infinitynon-parallel linesparallel linesMaster MOSIG Introduction to Projective Geometry

projective transformations that transform points into points and lines into lines and preserve the cross ratio (the collineations). In 1872,Felix Klein proposes the Erlangen program, at the Erlangen university, within which a geometry is not dened by the objects it represents but by their trans- formations, hence the study of invariants for a group of transformations. This yields a hierarchy of geometries, dened as groups of transformations, where the Euclidean geometry is part of the ane geometry which is itself included into the projective geometry.Figure 1.3: The geometry hierarchy.

1.3 Bibliography

The books below served as references for these notes. They include computer vision books that present comprehensive chapters on projective geometry. J.G. Semple and G.T. Kneebone,Algebraic projective geometry, Clarendon

Press, 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, MIT

Press (2001)

D. Forsyth and J. Ponce,Computer Vision: A Modern Approach, Prentice

Hall (2003)

Grenoble Universities 4Projective GeometryAffine GeometryEuclidean GeometryMaster MOSIG Introduction to Projective Geometry

Chapter 2

Projective Spaces

In this chapter, formal denitions and properties of projective spaces are given, regardless of the dimension. Specic cases such as the line and the plane are studied in subsequent chapters.

2.1 Denitions

Consider the real vector spaceRn+1of dimensionn+ 1. Letvbe a non-zero element ofRn+1then the setRvof all vectorskv,k2Ris called a ray (cf. gure 2.1).Figure 2.1: The rayRvis the set of all non-zero vectorskvwith directionv. Denition 2.1(Geometric Denition)The real projective spacePn, of dimen- sionn, associated toRn+1is the set of rays ofRn+1. An element ofPnis called a point and a set of linearly independent (respectively dependent) points ofPn is dened by a set of linearly independent (respectively dependent) rays. Grenoble Universities 5kvMaster MOSIG Introduction to Projective Geometry Figure 2.2: The projective space associated toR3is called the projective plane P 2. Denition 2.2(Algebraic Denition)A point of a real projective spacePnis represented by a vector of real coordinatesX= [x0;:::;xn]t, at least one of which is non-zero. Thefxigs are called the projective or homogeneous coordi- nates and two vectorsXandYrepresent the same point when there exists a scalark2Rsuch that: x i=kyi8i; which we will denote by: XY: Hence the projective coordinates of a point are dened up to a scale factor and the correspondence between points and coordinate vectors is not one-to-one. Projective coordinates relate to a projective basis: Denition 2.3(Projective Basis)A projective basis is a set of(n+ 2)points ofPn, no(n+ 1)of which are linearly dependent. For example: 2 6

666666641

0 03 7

777777752

6

666666640

1 03 7

77777775:::2

6

666666640

0 13 7

777777752

6

666666641

1 13 7

77777775

A

0A1::: AnA

Grenoble Universities 6ABC

ABC RRRMaster MOSIG Introduction to Projective Geometry is the canonical basis where thefAigs are called the basis points andAthe unit point. The relationship between projective coordinates and a projective basis is as follows. Projective CoordinatesLetfA0;::;An;Agbe a basis ofPnwith associated raysRviandRvrespectively. Then for any pointAofPnwith an associated rayRv, its projective coordinates [x0;:::;xn]tare such that: v=x0v0+:::+xnvn; where the scales of the vectorsfvigs associated to thefAigs are given by: v=v0+:::+vn; which determines thefxigs up to a scale factor.Note on projective coordinates To better understand the above characterization of the projective coor- dinates, let us consider any (n+ 1) vectorsviassociated to thefAigs. By denition they form a basis ofRn+1and any vectorvin this space can be uniquely decomposed as: v=u0v0+:::+unvn; ui2R8i: Thusvis determined by a single set of coordinatesfuigin the vector basisfvig. However the above unique decomposition with theui's does not transfer to the associated pointsAandfAigs ofPnsince the corre- spondence between points inPnand vectors inRn+1is not one-to-one. For instance replacing in the decompositionu0byu0=2 andv0by 2v0 still relatesAwith thefAigs but with a dierent set offuigs. In order to uniquely determine the decomposition, let us consider the additional pointAand letvbe one of its associated vector inRn+1then: v =u0v0+:::+unvn=v

0+:::+v

n:

The abovescaledvectorsv

iare well dened as soon asui6= 0;8i(true by the fact that, by denition,Ais linearly independent of any subset ofnpointsAi). Then, any vectorvassociated toAwrites: v=x0v

0+:::+xnv

n; where thefxigs can vary only by a global scale factor function of the scales ofvandv.Grenoble Universities 7Master MOSIG Introduction to Projective Geometry Denition 2.4(Projective Transformations)A matrixMof dimensions(n+

1)(n+1)such thatdet(M)6= 0, or equivalently non-singular, denes a linear

transformation fromPnto itself that is called a homography, a collineation or a projective transformation. Projective transformations are the most general transformations that pre- serve incidence relationships, i.e. collinearity and concurrence.

2.2 Properties

Some classical and fundamental properties of projective spaces folllow. Theorem 2.1Considermpoints ofPnthat are linearly independent with m < n. The set of points inPnthat are linearly dependent on thesempoints form a projective space of dimensionm1. When this dimension is equal to1,

2andn1, this space is called line, plane and hyperplane respectively. The set

of subspaces ofPnwith the same dimension is also a projective space. ExamplesLines are hyperplanes ofP2and they form a projective space of dimension 2. Theorem 2.2(Duality)The set of hyperplanes of a projective spacePnis a projective space of dimensionn. Any denition, property or theorem that applies

to the points of a projective space is also valid for its hyperplanes.Figure 2.3: Lines and points are dual inP2.

ExamplesPoints and lines are dual in the projective plane, 2points dene a lineis dual to 2lines dene a point. Another interesting illustration of the duality is the Desargues' theorem (see gure 2.4) that writes: If2triangles are such that the lines joining their corresponding vertices are concurrent then the points of intersections of the corresponding edges are

Grenoble Universities 82 lines define a point2 points define a lineMaster MOSIG Introduction to Projective Geometry

collinear, and which reciprocal is its dual (replace in the statementlines joiningwith points of intersections of,verticeswithedgesandconcurrentwithcollinearand vice versa).Figure 2.4: Desargues' theorem illustrated with parallel lines (hence concurrent lines in the projective sense) joining the corresponding vertices on 2 triangles. Theorem 2.3(Change of Basis)LetfX0;:::;Xn+1gbe a basis ofPn, i.e. no (n+ 1)of them are linearly dependent. IffA0;:::;An+1=Agis the canonical basis then there exists a non-singular matrixMof dimension(n+ 1)(n+ 1) such that:

MAi=kiXi;ki2R8i;

or equivalently:

MAiXi8i:

2matricesMandM0that satisfy this property dier by a non-zero scalar factor

only, which we will denote using the same notation:MM0. Grenoble Universities 9Master MOSIG Introduction to Projective Geometry Proof The matrixMsatises:MAi=kiXi8i. StackingA0;:::;Aninto a matrix we get, by denition of the canonical basis: [A0:::An] =In+1 hence:

M[A0:::An] =M= [k0X0:::knXN];

which determinesMup to the scale factorsfkig. Using this expression withAn+1:

MAn+1= [k0X0:::knXN]2

6

6666641

13 7

777775= [X0:::XN]2

6

666664k

0 k n3 7

777775;

and since:MAn+1=kn+1Xn+1we get: [X0:::XN]2 6

666664k

0 k n3 7

777775=kn+1Xn+1;

that gives thefkigs up to a single scale factor. Note also that thefkigs are necessarily non-zero otherwise (n+1) vectorsXiare linearly depen- dent by the above expression.Thus any basis ofPnis related to the canonical basis by a homography. A consequence of theorem 2.3 is that: Corollary 2.4LetfX0;:::;Xn+1gandfY0;:::;Yn+1gbe2basis ofPn, then there exists a non-singular matrixMof dimension(n+ 1)(n+ 1)such that:

MXiYi8i;

whereMis determined up to a scale factor. Grenoble Universities 10Master MOSIG Introduction to Projective Geometry Proof

By theorem 2.3:

LAi=kiXi;8i;

QAi=liYi;8i:

Thus: QL

1Xi=lik

iYi;8i; and the matrixM=QL1is therefore such that:

MXiYi;8i;

Now if there is a matrixM0such that:M0XiYi;8i, then replacing X iwe get: M

0LAiYi;8i;

and by theorem 2.3:M0LML, henceM0M.Figure 2.5: Change of basis inP2or projective transformation betweenA;B;C

andA0;B0;C0. Figure 2.5 illustrates the change of basis inP2.A;B;CandA0;B0;C0are 2 dierent representations of the same rays and are thus related by a homography (projective transformation). Note that this gure illustrates also the relationship between coplanar points and their images by a perspective projection. Grenoble Universities 11ABCA'B'C'Master MOSIG Introduction to Projective Geometry

2.3 The hyperplane at innity

Figure 2.6: InP2any lineLis the hyperplane at innity for the ane space P

2nL. In this ane space, all lines that share the same direction are concurrent

on the line at innity. The projective spacePncan also be seen as the completion of a hyperplane HofPnand the set complementAn=PnnH.Anis then the ane space of dimensionn(associated to the vector spaceRn) andHis its hyperplane at innity also called ideal hyperplane. This terminology is used sinceHis the locus of points inPnwhere parallel lines ofAnintersect. As an example, assume that [x0;:::;xn]tare the homogeneous coordinates of points inPnand consider the ane spaceAnof points with inhomoge- neous coordinates (not dened up to scale factor) [x0;:::;xn1]t. Then the lo- cus of points inPnthat are not reachable withinAnis the hyperplane with equationxn= 0. To understand this, observe that there is a one-to-one mapping between points inAnand points inPnwith homogeneous coordi- nates [x0;:::;xn1;1]t. Going along the direction [x0;:::;xn1]tinAnby chang- ing the value ofkin [x0=k;:::;xn1=k;1]twe see that there is a point at the limitk!0, that is at innity and not inAn, with homogeneous coordinates [x0=k;:::;xn1=k;1]t[x0;:::;xn1;k]t=k!0[x0;:::;xn1;0]t(see Figure 2.6). This point belongs to the hyperplane at innity (or the ideal hyperplane) asso- ciated withAn. Any hyperplaneHofPnis thus the plane at innity of the ane spacePnn H. Reciprocally, adding to anyn-dimensional ane spaceAnthe hyperplane of its points at innity converts it into a projective space of dimensionn. This is called the projective completion ofAn.

Grenoble Universities 12∞

[X /k,X /k,1]01[X ,X ,0]01k α 0Master MOSIG Introduction to Projective Geometry

Chapter 3

The projective line

The spaceP1is called the projective line. It is the completion of the ane line with a particular projective point, the point at innity, as will be further detailed in this chapter. The projective line is useful to introduce projective notions, such as the cross-ratio, in a simple and intuitive way.

3.1 Introduction

The canonical basis ofP1is:

A 0=1 0 ; A 1=0 1 ; A =A1+A2=1 1 A point ofP1is represented by a vector of 2 homogeneous coordinatesX [x0;x1]6= [0;0]. Hence:Xx0A0+x1A1. Now consider the subspace ofP1such thatx16= 0. This is equivalent to exclude the pointA0and it denes the ane lineA1. Point onA1can be described by a single parameterksuch that:

X=kA0+A1;

wherek=x0=x1is the ane coordinate.Figure 3.1: On the ane line, the coordinatekofCin the coordinate frame

[O;B] isk=OC=OB. A

0is the point at innity, or ideal point, for the ane spaceP1nA0.

Grenoble Universities 13∞BCO∞Master MOSIG Introduction to Projective Geometry

3.2 Projective transformation ofP1

A projective transformation ofP1is represented by a 22 non singular matrix

Hdened up to a scale factor:

Hh1h2 h 3h4 The above matrix has 3 degrees of freedom since it is dened up to a scale factor. From corollary 2.3 of section 2.2, it follows that 3 point correspondences,

or equivalently 2 basis ofP1, are required to estimateH.Figure 3.2: A homography inP1is dened by 3 point correspondences, con-

cerning possibly the innite point. The restriction of the projective transformationHto the ane spaceA1is a transformationMthat does not aect the point at innity, i.e.MA0A0.

HenceMis of the form:

Mm1m2 0 1 wherem1is a scale factor andm2a translation parameter. 2 point correspon- dences are sucient to estimateM.

3.3 The cross-ratio

The cross-ratio, also called double ratio ("bi-rapport" in French), is the funda- mental invariant ofP1, that is to say a quantity that is preserved by projective transformation. It is the projective equivalent to the Euclidean distance with rigid transformations. LetA;B;C;Dbe 4 points on the projective line then their cross-ratio writes: fA;B;C;Dg=jACjBDjjBCjADj; wherejABj= detxA0xB0xA1xB1 = (xA0xB1)(xA1xB0).

Some remarks are in order:

Grenoble Universities 14H∞Master MOSIG Introduction to Projective Geometry Figure 3.3:A;B;C;DandA0;B0;C0;D0are related by a projective transforma- tion, hence their cross-ratios are equal. 1. The cross-ratio is indep endentof the basis c hosenfor P1. 2. On the ane line, A= [ka;1],B= [kb;1],etc. and the cross-ratio be- comes: fA;B;C;Dg=(kAkC)(kBkD)(kBkC)(kAkD); hence in the Euclidean space: fA;B;C;Dg=dACdBDd

BCdAD;

withdfgbeing the Euclidean distance between 2 points. 3. f1;B;C;Dg=(kBkD)(kBkC); fA;1;C;Dg=(kAkC)(kAkD); 4. By p ermutingthe p ointsA;B;C;D, 24 quadruplets can be formed. These quadruplets dene only 6 dierent values of the cross-ratio:;1 ;1quotesdbs_dbs43.pdfusesText_43

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