Chapter 5 Basics of Projective Geometry

1 Chapter 5 Basics of Projective GeometryThink geometrically, prove algebraically. John Why Projective Spaces?For a novice, Projective Geometry usually appears to be a bitodd, and it is notobvious to motivate why its introduction is inevitable and motivations arises from algebraic main goal of algebraic Geometry is to study the propertiesofgeometricob-jects, such as curves and surfaces, defined implicitly in terms of algebraic instance, the equationx2+y2 1=0defines a circle ,wecanconsiderthecurvesdefinedbygenerale quationsax2+by2+cxy+dx+ey+f=0of degree 2, known curves according to their generic geometric shape. This is indeed for so-called singular cases, we get ellipses, parabolas, and hyperbolas. Thesame question can be asked for surfaces defined by quadratic equations, knownasquadrics,andagain, ,theseclassificationsare a bit artificial. For example, an ellipse and a hyperbola differ by the fact thatahyperbolahaspointsatinfinity,andyet ,theirgeometricproperties are identical,provided that points at infinity are handled important problem is the study of intersection of geometric objects (de-fined algebraically).

2 For example, given two curvesC1andC2of degreemandn,respectively, what is the number of intersection points ofC1andC2?(bydegreeofthe curve we mean the total degree of the defining polynomial).1031045 BasicsofProjectiveGeometryWell, it depends! Even in the case of lines (whenm=n=1), there are threepossibilities: either the lines coincide, or they are parallel, or there is a single inter-section point. In general, we expectmnintersection points, but some of these pointsmay be missing because they are at infinity, because they coincide, or because theyare begins to transpire is that points at infinity cause trouble. They cause ex-ceptions that invalidate geometric theorems (for example,consider the more generalversions of the theorems of Pappus and Desargues from ),andmakeitdifficult to classify geometric objects. Projective Geometry is designed to deal with points at infinity and regular points in a uniform way, without making a distinc-tion.

3 Points at infinity are now just ordinary points, and manythingsbecomesim-pler. For example, the classification of conics and quadricsbecomes simpler, andintersection theory becomes cleaner (although, to be honest, we need to considercomplex Projective spaces).Technically, Projective Geometry can be defined axiomatically, or by buidlingupon linear algebra. Historically, the axiomatic approachcame first (see Veblen andYoung [28, 29], Emil Artin [1], and Coxeter [7, 8, 5, 6]). Although very beautiful andelegant, we believe that it is a harder approach than the linear algebraic approach. Inthe linear algebraic approach, all notions are considered , of coordinates, thiscorresponds to homogenizing. For example, the homogeneous equation of a conicisax2+by2+cxy+dxz+eyz+fz2= , regular points are points of coordinates(x,y,z)withz"=0, and points at infinityare points of coordinates(x,y,0)(withx,y,znot all null, and up to a scalar).

4 There isausefulmodel(interpretation)ofplanepro jectivegeometry in terms of the centralprojection inR3from the origin onto the planez=1. Another useful model is thespherical (or the half-spherical) model. In the spherical model, a Projective pointcorresponds to a pair of antipodal points on the affine Geometry is the study of properties invariant underaffine bijections, Projective Geometry is the study of properties invariant under bijective projectivemaps. Roughly speaking, Projective maps are linear maps up our presentation of affine Geometry , we will define Projective spaces, projectivesubspaces, Projective frames, and Projective maps. The analogy will fade away whenwe define the Projective completion of an affine space, and of the virtues of Projective Geometry is that it yields a very clean presentationof rational curves and rational surfaces. The general idea isthataplanerationalcurve is the projection of a simpler curve in a larger space, apolynomial curve inR3,ontotheplanez=1, as we now curves are curves defined parametrically in termsofpolynomi-als.

5 More specifically, ifEis an affine space of finite dimensionn 2and(a0,(e1,..,en))is an affine frame forE,apolynomialcurveofdegreemis a mapF:A Esuch thatF(t)=a0+F1(t)e1+ +Fn(t)en, Why Projective Spaces?105for allt A,whereF1(t),..,Fn(t)are polynomials of degree at many curves can be defined, it is somewhat embarassing that a circlecannot be defined in such a way. In fact, many interesting curves cannot be definedthis way, for example, ellipses and hyperbolas. A rather simple way to extend theclass of curves defined parametrically is to allow rational functions instead of poly-nomials. Aparametric rational curveof degreemis a functionF:A EsuchthatF(t)=a0+F1(t)Fn+1(t)e1+ +Fn(t)Fn+1(t)en,for allt A,whereF1(t),..,Fn(t),Fn+1(t)are polynomials of degree at example, a circle inA2can be defined by the rational mapF(t)=a0+1 t21+t2e1+2t1+ the above example, the denominatorF3(t)=1+t2never takes the value 0whentranges overA,butconsiderthefollowingcurveinA2:G (t)=a0+t2te1+ thatG(0)is undefined.

6 The curve defined above is a hyperbola, and fortclose to 0, the point on the curve goes toward infinity in one ofthe two is to workin a Projective space. Intuitively, this means viewing a rational curve inAnas someappropriate projection of a polynomial curve inAn+1, an affine spaceE,foranyhyperplaneHinEand any pointa0not inH,thecentral projection (or conic projection, or perspective projection) of center a0ontoH,isthepartialmappdefined as follows: For every pointxnot in the hyperplanepassing througha0and parallel toH,wedefinep(x)as the intersection of the linedefined bya0andxwith the example, we can viewGas a rational curve inA3given byG1(t)=a0+t2e1+e2+ we project this curveG1(in fact, a parabola inA3)usingthecentralprojection(perspecti ve projection) of centera0onto the plane of equationx3=1, we get theprevious hyperbola. Fort=0, the pointG1(0)=a0+e2inA3is in the plane ofequationx3=0, and its projection is undefined.

7 We can consider thatG1(0)=a0+e2inA3is projected to infinity in the direction ofe2in the planex3=0. In the settingof Projective spaces, this direction corresponds rigorously to a point at us verify that the central projection used in the previousexamplehasthede-sired effect. Let us assume thatEhas dimensionn+1andthat(a0,(e1,..,en+1))is an affine frame (x)of a pointx Eonto the hyperplaneHof equationxn+1=1(thecenterof1065 BasicsofProjectiveGeometryprojection beinga0). Ifx=a0+x1e1+ +xnen+xn+1en+1,assuming thatxn+1"=0; a point on the line passing througha0andxhas coordinatesof the form( x1,.., xn+1);andp(x),thecentralprojectionofxont o the hyper-planeHof equationxn+1=1, is the intersection of the line froma0toxand xn+1=1, and the coordinates ofp(x)are(x1xn+1,..,xnxn+1,1).Note thatp(x)is undefined whenxn+1=0. In Projective spaces, we can make senseof such above calculation confirms thatG(t)is a central projection ofG1(t).

8 Simi-larly, if we define the curveF1inA3byF1(t)=a0+(1 t2)e1+2te2+(1+t2)e3,the central projection of the polynomial curveF1(again, a parabola inA3)ontotheplane of equationx3= we just sketched is a general method to deal with our hat construction to embed an affine spaceEinto a vector space Ehavingone more dimension, then construct the Projective spaceP( E).Thisturnsouttobe the Projective completion of the affine inP( E),basicallyasthecentralprojectionofapol ynomialcurvein EbackontoP( E). lack of space, such a presentation is omitted from the ,itcan be found in the additional material on the web site; jean/ generally, the Projective completion of an affine spaceis a very convenienttool to handle points at infinity in a clean Chapter contains a brief presentation of concepts of Projective following concepts are presented: Projective spaces, Projective frames, homo-geneous coordinates, Projective maps, Projective hyperplanes, multiprojective maps,affine patches.

9 The Projective completion of an affine space ispresentedusingthe hat construction. The theorems of Pappus and Desargues areproved,usingthemethod in which points are sent to infinity. We also discussthe cross-ratio andduality. The Chapter ends with a very brief explanation of theuseofthecomplexifi-cation of a Projective space in order to define the notion of angle and orthogonalityin a Projective setting. We also include a short section on applications of projectivegeometry, notably to computer vision (camera calibration),efficientcommunication,and error-correcting Projective Projective SpacesAs in the case of affine Geometry , our presentation of Projective Geometry is rathersketchy and biased toward the algorithmic Geometry of treatment of Projective Geometry , we recommendBerger [3, 4], Samuel[23], Pedoe [21], Coxeter [7, 8, 5, 6], Beutelspacher and Rosenbaum [2], Fres-nel [14], Sidler [24], Tisseron [26], Lehmann and Bkouche [20], vienne [30],and the classical treatise by Veblen and Young [28, 29], which, although slightlyold-fashioned, is definitely worth reading.

10 Emil Artin s famous book [1] contains,among other things, an axiomatic presentation of projectivegeometry,andawealthof geometric material presented from an algebraic point of view. Other oldies butgoodies include the beautiful books by Darboux [9] and Klein[19].Foradevel-opment of Projective Geometry addressing the delicate problem of orientation, seeStolfi [25], and for an approach geared towards computer graphics, see Penna andPatterson [22].First, we define Projective spaces, allowing the fieldKto be arbitrary (whichdoes no harm, and is needed to allow finite and complex Projective spaces). Roughlyspeaking, every Projective concept is a linea algebraic concept up to a scalar. Forspaces, this is made precise as followsDefinition a vector spaceEover a fieldK,theprojective spaceP(E)induced by Eis the set(E {0})/ of equivalence classes of nonzero vectors inEunder the equivalence relation defined such that for allu,v E {0},u viffv= u,for some K {0}.