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Chapter5
Basics of Projective Geometry
Think geometrically, prove algebraically.
—JohnTate
5.1 WhyProjective Spaces?
For a novice, projective geometry usually appears to be a bit odd, and it is not
obvioustomotivatewhyitsintroductionisinevitableandinfactfruitful.Oneofthe
mainmotivationsarises from algebraic geometry.
Themaingoalofalgebraic geometry is to study the propertiesofgeometricob-
jects, such as curvesand surfaces,definedimplicitly in termsofalgebraicequations.
For instance, the equation
x2+y2!1=0
definesacircleinR2.Moregenerally,wecanconsiderthecurvesdefinedbygeneral
equations
2 2
ax +by +cxy+dx+ey+f =0
ofdegree2,knownasconics.Itisthennaturaltoaskwhetheritispossibletoclassify
these curves according to their generic geometric shape. This is indeed possible.
Except for so-called singular cases, we get ellipses, parabolas, and hyperbolas. The
same question can be asked for surfaces defined by quadratic equations, known
as quadrics,andagain,aclassificationispossible.However,theseclassifications
are a bit artificial. For example, an ellipse and a hyperbola differ by the fact that
ahyperbolahaspointsatinfinity,andyet,theirgeometricproperties are identical,
providedthat points at infinity are handled properly.
Anotherimportantproblemis the study of intersection of geometric objects (de-
fined algebraically). For example, given two curves C and C of degree m and n,
1 2
respectively, what is the number of intersection points of C and C ?(bydegreeof
1 2
the curve we mean the total degree of the defining polynomial).
103
104 5BasicsofProjectiveGeometry
Well, it depends! Even in the case of lines (when m = n = 1), there are three
possibilities: either the lines coincide, or they are parallel, or there is a single inter-
section point. In general, we expect mn intersection points, but some of these points
maybemissingbecausetheyareat infinity, because they coincide, or because they
are imaginary.
Whatbeginstotranspire is that “points at infinity” cause trouble. They cause ex-
ceptionsthat invalidategeometrictheorems(forexample,considerthemoregeneral
versions of the theorems of Pappus and Desargues from Section2.12),andmakeit
difficult 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. Points at infinity are now just ordinary points, and manythingsbecomesim-
pler. For example, the classification of conics and quadrics becomes simpler, and
intersection theory becomes cleaner (although, to be honest, we need to consider
complexprojective spaces).
Technically, projective geometry can be defined axiomatically, or by buidling
uponlinearalgebra.Historically, the axiomatic approachcame first (see Veblen and
Young [28, 29], Emil Artin [1], and Coxeter [7, 8, 5, 6]). Althoughverybeautifuland
elegant, we believe that it is a harder approach than the linear algebraic approach. In
the linear algebraic approach, all notions are considered uptoascalar.Forexample,
aprojectivepointisreallyalinethroughtheorigin.Interms of coordinates, this
correspondsto“homogenizing.”Forexample,thehomogeneousequationofaconic
is
2 2 2
ax +by +cxy+dxz+eyz+fz =0.
Now,regularpointsarepointsofcoordinates(x,y,z)withz"=0,andpointsatinfinity
are pointsofcoordinates(x,y,0) (withx,y,znotallnull,anduptoascalar).Thereis
ausefulmodel(interpretation)ofplaneprojectivegeometry in terms of the central
projection in R3 from the origin onto the plane z = 1. Another useful model is the
spherical (or the half-spherical) model. In the spherical model, a projective point
correspondsto a pair of antipodal points on the sphere.
As affine geometry is the study of properties invariant under affine bijections,
projective geometry is the study of properties invariant under bijective projective
maps. Roughlyspeaking,projective maps are linear maps up toascalar.Inanalogy
withourpresentationofaffinegeometry,wewilldefineprojectivespaces,projective
subspaces,projectiveframes,andprojectivemaps.Theanalogywillfadeawaywhen
wedefinetheprojectivecompletionof an affine space, and whenwedefineduality.
Oneofthevirtuesofprojectivegeometryisthatityieldsaverycleanpresentation
of rational curves and rational surfaces. The general idea isthataplanerational
curve is the projection of a simpler curve in a larger space, a polynomial curve in
R3,ontotheplanez=1,aswenowexplain.
Polynomial curves are curves defined parametrically in termsofpolynomi-
als. More specifically, if E is an affine space of finite dimension n # 2and
(a0,(e1,...,en)) is an affine frame for E,apolynomialcurveofdegreem is a map
F: A→E suchthat
F(t)=a +F(t)e +···+F (t)e ,
0 1 1 n n
5.1 WhyProjective Spaces? 105
for all t ∈ A,whereF (t),...,F (t) are polynomials of degree at most m.
1 n
Although many curves can be defined, it is somewhat embarassing that a circle
cannot be defined in such a way. In fact, many interesting curves cannot be defined
this way, for example, ellipses and hyperbolas. A rather simple way to extend the
class of curves defined parametrically is to allow rational functions instead of poly-
nomials. A parametric rational curve of degree m is a function F: A → E such
that
F (t) F (t)
F(t)=a + 1 e +···+ n e ,
0 F (t) 1 F (t) n
n+1 n+1
for all t ∈ A,whereF (t),...,F (t),F (t) are polynomials of degree at most m.
1 n n+1
For example, a circle in A2 can be defined by the rational map
1!t2 2t
F(t)=a0+1+t2e1+1+t2e2.
In the above example, the denominator F (t)=1+t2 never takes the value 0
3
whent rangesoverA,butconsiderthefollowingcurveinA2:
t2 1
G(t)=a0+ e1+ e2.
t t
Observe that G(0) is undefined. The curve defined above is a hyperbola, and for t
close to 0, the point on the curve goes toward infinity in one of the two asymptotic
directions.
Acleanwaytohandlethesituationinwhichthedenominatorvanishesistowork
in a projective space. Intuitively, this means viewing a rational curve in An as some
appropriate projection of a polynomial curve in An+1,backontoAn.
GivenanaffinespaceE,foranyhyperplaneH inE andanypointa0notinH,the
central projection (or conic projection, or perspective projection) of center a0 onto
H,isthepartialmapp defined as follows: For every point x not in the hyperplane
passing through a0 and parallel to H,wedefinep(x) as the intersection of the line
definedbya0 andxwiththehyperplaneH.
For example, we can view G as a rational curve in A3 given by
G (t)=a +t2e +e +te .
1 0 1 2 3
If we project this curve G (in fact, a parabola in A3)usingthecentralprojection
1
(perspective projection) of center a0 onto the plane of equation x3 = 1, we get the
previous hyperbola. For t = 0, the point G (0)=a +e in A3 is in the plane of
1 0 2
equation x3 =0, and its projection is undefined.We can consider that G1(0)=a0+
e inA3isprojectedtoinfinityinthedirectionofe intheplanex =0.Inthesetting
2 2 3
of projective spaces, this direction corresponds rigorously to a point at infinity.
Let us verify that the central projection used in the previousexamplehasthede-
sired effect. Let us assume that E has dimension n+1andthat(a0,(e1,...,en+1))
is an affine frame for E.Wewanttodeterminethecoordinatesofthecentralprojec-
tion p(x) of a point x ∈ E onto the hyperplaneH of equation xn+1 =1(thecenterof
106 5BasicsofProjectiveGeometry
projection being a0). If
x =a0+x1e1+···+xnen+xn+1en+1,
assumingthatxn+1"=0;apointonthelinepassingthrougha0 andxhascoordinates
of the form (λx1,...,λxn+1);andp(x),thecentralprojectionofx onto the hyper-
plane H of equation xn+1 = 1, is the intersection of the line from a0 to x and this
hyperplaneH.Thuswemusthaveλxn+1=1,andthecoordinatesof p(x)are
! x1 xn "
,..., ,1 .
xn+1 xn+1
Notethat p(x) is undefinedwhenxn+1 =0.Inprojectivespaces,wecanmakesense
of such points.
The above calculation confirms that G(t) is a central projection of G1(t).Simi-
larly, if we define the curve F in A3 by
1
F (t)=a +(1!t2)e +2te +(1+t2)e ,
1 0 1 2 3
the central projection of the polynomial curve F (again, a parabola in A3)ontothe
1
plane of equation x3 = 1isthecircleF.
What we just sketched is a general method to deal with rationalcurves.Wecan
#
use our “hat construction” to embed an affine space E into a vector space E having
$ %
#
one more dimension, then construct the projective space P E .Thisturnsoutto
be the “projective completion” of the affine space E.Thenwecandefinearational
$ %
# #
curve in P E ,basicallyasthecentralprojectionofapolynomialcurvein E back
$ %
#
onto P E .Thesameapproachcanbeusedtodealwithrationalsurfaces.Dueto
the lack of space, such a presentation is omitted from the maintext.However,it
can be found in the additional material on the web site; see http://www.cis.
upenn.edu/ jean/gbooks/geom2.html.
˜
Moregenerally,theprojectivecompletionof an affinespace is a very convenient
tool to handle “points at infinity” in a clean fashion.
This chapter contains a brief presentation of concepts of projective geometry.
The following concepts are presented: projective spaces, projective frames, homo-
geneouscoordinates,projectivemaps,projectivehyperplanes,multiprojectivemaps,
affine patches. The projective completion of an affine space ispresentedusingthe
“hat construction.” The theorems of Pappus and Desargues areproved,usingthe
method in which points are “sent to infinity.” We also discuss the cross-ratio and
duality. The chapter ends with a very brief explanation of theuseofthecomplexifi-
cation of a projective space in order to define the notion of angle and orthogonality
in a projective setting. We also include a short section on applications of projective
geometry,notablytocomputervision(cameracalibration),efficientcommunication,
and error-correctingcodes.
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