Chapter 11 Duality and Polarity

Home , Duality (mathematics), Incidence (geometry), Projective plane, Real projective plane

Chapter 11

Duality and Polarity Math 4520 Fall 2017

In previous sections we have seen the definition of an abstract projective plane, what it means for two planes to be equivalent, and at least two examples of seemingly different but equivalent projective planes, the extended Euclidean plane and the definition using homogeneous coordinates. Here we review some of these “models” for the projective plane and use those descriptions to define a “polarity”, which is a kind of self-equivalence that interchanges points and lines.

11.1 Models of the projective plane

Recall, for the definition of a projective plane, we did not require that incidence of a point and line necessarily be such that the point was an element of the line. If we want to insist, we could redefine all the lines so that each line is the collection of points that are incident to it, but in some models, that will come up shortly, it is convenient to allow ourselves the freedom of a more general incidence relation. Recall that our first model of the projective plane was the Extended Euclidean Plane. Once we created our ideal points and line at infinity, incidence was defined as “being a member of”. In Chapter 9 we defined another model for “the” projective plane using homogeneous coordinates in some field. A projective point was defined as a line through the origin in the 3-dimensional vector space over the field. A projective line was defined as a plane in the vector space. Incidence was defined as set containment. Let us call this model the Homogeneous Model 1. We can alter this definition slightly and say that a projective point is the set of vectors on a line through the origin, except for the origin itself. We say that a projective line is a line through the origin with the origin removed. The line in the vector space corresponds to the [A, B, C] vector that makes up the coefficients of the equation that defines the plane in the Homogeneous Model 1. Thus incidence between a projective point and a .projective line is perpendicularity between the corresponding lines in the vector space. This definition has the advantage that points and lines are treated more equally, and the language of equivalence classes on the non-zero vectors in the 3-dimensional vector space can be used. We call this model the Homogeneous Model 2.

1 2 CHAPTER 11. DUALITY AND POLARITY MATH 4520 FALL 2017

In order to get a better understanding of more of the geometry than the incidence struc- ture, we can alter the above models even further in the case of real projective geometry. We can intersect the unit sphere in Euclidean 3-space with each of the classes in the Homoge- neous models 1 and 2. We say that two points p and q on the unit sphere in real 3-space are antipodal2 if q = −p. So two distinctCLASSICAL points GEOMETRIES on the unit sphere are antipodal if and only if they lie on a line through the origin. We say that a great circle is the intersection of a plane throughand only the if origin they lie with on a the line unit through sphere. the origin. We say that a great circle is the intersection Weof a now plane deﬁne through the the Sphere origin Model with the 1 forunit the sphere. real projective plane. Here a projective point We now define the Sphere Model 1 for the real projective plane. Here a projective point is ais pair a pair of antipodalof antipodal points points on on thethe realreal unit unit sphere. sphere. A Aprojective projective line lineis a great is a greatcircle circleon on the unitthe unit sphere. sphere. Incidence Incidence is is just just setset containment. Clearly Clearly this this model model is equivalent is equivalent to the to the HomogeneousHomogeneous2 ModelModel1 1 over over the the real realCLASSICAL ﬁeld. field. SeeSeeGEOMETRIES FigureFigure 12.1.1 11.1

and only if they lie on a line through the origin. We say that a great circle is the intersection of a plane through the origin with the unit sphere. We now define the Sphere Model 1 for the real projective plane. Here a projective point is a pair of antipodal points on the real unit sphere. A projective line is a great circle on the unit sphere. Incidence is just set containment. Clearly this model is equivalent to the HomogeneousModel1 over the real field. SeeFigure 12.1.1

Figure 12.1.1 Figure 11.1 Finally v.'edefine the Sphere Model 2 for the real projective plane. Again a projective point is defined to be a pair of antipodalFigure points. 12.1.1 However, a projective line is defined to Finallybe a pair we of define Finallyantipodal v.'e thedefine points. Sphere the Sphere Incidence Model Model 2 2is for perpendicularity. the the real real projective projective plane. Again plane. a projective Again a projective point isWe defined summarizepoint to is bedefined this a pair into betheof a pairfollowing antipodal of antipodal table. points. points. However, However, a projective a projective line is defined line to is defined to be be a pair of antipodal points. Incidence is perpendicularity. a pair of antipodalWe summarize points. this Incidence in the following is perpendicularity. table. We summarize this in Figure 11.2.

The Plane Points Lines Incidence The Plane Points Lines Incidence Extended Finite points Ordinary points Extended Finite points Ordinary points Euclidean EuclideanPlane Plane & ideal& ideal points points atat && the the line line at at "is a member"is a memberof" of" infinityinfinity infinityinfinity

Homogeneous Lines through Planes through Set containment Homogeneous Lines through Planes through Set containment coordinates 1 O O coordinates 1 O O Homogeneous Lines through O Lines through O Perpendicularity

coordinates 2 except O except 0 Homogeneous Lines through O Lines through O Perpendicularity coordinates 2 exceptAntipodal O Greatexcept circles on 0 Sphere model 1 points on the the sphere Set contajnment Antipodalsphere Great circles on Sphere model 1 pointsAntipodal on the Antipodalthe sphere Set contajnment Sphere model 2 spherepoints on the points on the Perpendicularity sphere sphere Antipodal Antipodal Sphere model 2 points on the points on the Perpendicularity sphere Figure 11.2 sphere

It is a good exercise to convince yourself that all of these models for the real projective plane are equivalent. 11.2. POLARITIES DUALITY AND POLARITY 3 3

11.2 PolaritiesIt is a good exercise to convince yourself that all of these models for the real projective plane are equivalent. Recall that our axioms for a projective plane had a symmetry with respect points and lines. In fact, the12.2 word Polarities “point” and the word “line” can be interchanged and the axioms remain the same. AnyRecall statement that our axioms in projective for a projective geometry plane had can a symmetry reverse thewith wordsrespect points “line” and and “point” lines. In fact, the word "point" and the word "line" can be interchanged and the axioms and it shouldremain make the same. sense. Any But statement will one in projective statement geometry be true can whenreverse the the words other "line" is? Ifand the truth of one statement"point" follows and it should directly make from sense. the But original will one threestatement axioms, be true thewhen answer the other will is? beIf yes. But some of thethe statements truth of one statement that we follows proved, directly for examplefrom the original Pascal’s three Theorem, axioms, the relied answer on will using more be yes. But some of the statements that we proved, for example Pascal's Theorem, relied than our threeon using axioms. more than So our we three look axioms. for a more So we explicitlook for a way more of explicit relating way of the relating points the and lines. points and lines.

Figure 11.3 Recall from Chapter 11 that a collineation between projective planes was an incidence preserving function taking points to points and lines to lines. Here we define a similar notion but with points and lines being interchanged. Let 71"be any projective plane and let

Recall from Chapter 11 that a0: : collineation {points of 7!"} -+ between {lines of 7!" projective} planes was an incidence preserving function taking points to points and lines to lines. Here we deﬁne a similar notion but with pointsbe a one-to-one and lines onto being function interchanged. suchthat for any Let pointπ bep of any 7!' and projective any line 1 plane of 7!', p andand let1 are incident if and only if o.(p) and 0.(1) are incident, where we use a to denote the inverse of a. Such a function is called a polarity. For a point p we call o.(p) the polar of p, and for a line I we call 0.(1) theα :pole{points of I. Note of πthat} → the { linesexistence of πof} such a polarity says that, in a certain sense, points and lines are interchangeable. We present a few examples. Let us take the Homogeneous Coordinates Model I. We can be a one-to-onetake the onto polar functionof a line through such thatthe origin for anyto be pointthe planep throughof π and the anyorigin line perpendicularl of π, p and l are incident ifto and that only line, ifandα (sop )the and pole αof( lplane) are is incident,the line through where thewe origin use perpendicularα to denote to it. the inverse of For the Sphere Model I, the polar of a pair of antipodal points is the great circle α. (This isequidistant no problem, to them, since and thethe pole domain of a great and circle range is the of pairα are of antipodal diﬀerent. points We that think are of α as a correspondenceon the betweenline through the the setorigin of perpendicular points and to the the setplane of of lines.)the great Suchcircle. aSo function the pole of is called a polarity. For a point p we call α(p) the polar of p, and for a line l we call α(l) the pole of l. Note that the existence of such a polarity allows us to interchange points and lines in a very precise way. We present a few examples. Let us take the Homogeneous Coordinates Model 1. We can take the polar of a line through the origin to be the plane through the origin perpendicular to that line, and so the pole of plane is the line through the origin perpendicular to it. For the Sphere Model 1, the polar of a pair of antipodal points is the great circle equidis- tant to them, and the pole of a great circle is the pair of antipodal points that are on the line through the origin perpendicular to the plane of the great circle. So the pole of the equator on our Earth are the North and South Poles, which is the reason for its name. See Figure 11.4. For the Homogeneous Model 2 and the Sphere Model 2, the polarity takes on a very simple form. Recall that points in the projective plane were equivalence classes of non-zero 4 CHAPTER 11. DUALITY AND POLARITY MATH 4520 FALL 2017

great circle

-p

Figure 11.4

column vectors and lines were non-zero row vectors. The polarity is given as follows:

x α y = x, y, z . z

It is easy to check that this is a polarity and is equivalent to the other polarities mentioned before, where the equivalence is in terms of the natural collineation correspondences that were described above.

11.3 More polarities

One unfortunate (or possibly just distinctive) property of the polarity mentioned above is that a projective point is never incident to its polar. This kind of polarity is called an elliptic polarity. Otherwise the polarity is called hyperbolic. Note that a polarity followed by a collineation is again a polarity. Thus the following function α is seen to be a polarity.

x αy = x, y, −z . z

Indeed, a point is incident to its polar if and only if

xx x 2 2 2 0 = αyy = x, y, −z y = x + y − z . z z z

So a point is incident to its polar if and only if it lies on a conic, the circle x2 + y2 = 1 in the Aﬃne plane z = 1. Thus α is a hyperbolic polarity. For this polarity we now give a very simple geometric description using the circle C deﬁned above. First, we claim that for any point p in C, α(p) is tangent to C. If not, then α(p) is incident to another point p0 in C as in Figure 11.5. Then α(α(p)) = p is incident to α(p0); DUALITY AND POLARITY 5 11.3. MORE POLARITIES DUALITY AND POLARITY 5 5

For this polarity we now give a very simple geometric description using the circle C defined above. First, \\'e claim that for any point pFigure in C, a(p) 11.5 is tangent to C. If not, then a(p) is incident to another point p. in C as in Figure 12.3.1. Then a( a(p ) ) = p is incident to For this polarity we now give a very simple geometric description using the circle C a(p.); and since p. is on C, a(p.) is incident to p. as well. Thus a(p.) = a(p), which definedcontradicts above. that a is one-to-one. Thus a(p) is tangent to C . First, \\'e claim that for any point p in C, a(p) is tangent to C. If not, then a(p) is incident to another point p. in C as in Figure 12.3.1. Then a( a(p ) ) = p is incident to a(p.); and since p. is on C, a(p.) is incident to p. as well. Thus a(p.) = a(p), which contradicts that a is one-to-one. Thus a(p) is tangent to C .

Figure 12.3.2 Figure 11.6 Supposethat P is a point outside the circle G. Let TI and T2 be the two tangents to G that are incident to p. See"Figure 12.3.2. So Q(T1) and Q(T2) are the points of tangency of TI and T2, respectively, on G, and Q(T1) and Q(T2) are both incident to Q(p). Thus Q(p) is the line determined by the two points of tangency of TI and T2. and since Supposep0 is on thatC, Pα (isp 0a) point is incident inside theFigure to circlep0 as G.12.3.2 well. Choose Thus two αdistinct(p0) =linesα( pII), and which 12 that contradicts that α isare one-to-one. incident to p. Thus Reverseα(p the) is construction tangent to ofC Figure. 12.3.2 to find the poles PI = Q(11) Supposeand P2that = 0.(12). P is Thena point PI andoutside P2 are the incident circle G.to Q(pLet ), TIand and so theyT2 be determine the two thetangents polar to G Suppose that p is a point outside the circle C. Let T and T be the two tangents to C that areQ(p ),incident and so theyto p. determine See"Figure the polar 12.3.2. Q(p ).So See Q(T1)Figure and 12.3.3. 1Q(T2) are2 the points of tangency We should keep in mind that the above construction must be correct becausewe know thatof areTI and incident T2, respectively, to p. See Figure on G, 11.6. and Q(T1) So α(T and1) and Q(T2)α(T are2) areboth the incident points to of tangencyQ(p). Thus of T1 and T ,from respectively, Section 12.2 on thatC, andsuch αa (hyperbolicT ) and αpolarity(T ) are exists. both It incidentturns out tothatα (wep). could Thus haveα(p) is the Q(p)2 ischosen the lineany conicdetermined to replace by G the in 1the two above points construction.2 of tangency of TI and T2. line determinedSuppose that by P the is twoa point points inside of tangencythe circle ofG. TChoose1 and T two2. distinct lines II and 12 that areSuppose incident that to pp. isReverse a point the inside construction the circle of CFigure. Choose 12.3.2 two to distinctfind the linespoles lPI1 and= Q(11)l2 that areand incident P2 = to0.(12).p. Reverse Then PI the and construction P2 are incident of Figure to Q(p 11.6 ), and to so ﬁnd they the determine poles p1 the= α polar(l1) and p2 Q(p= α ),(l 2and). Then so theyp1 determineand p2 are the incident polar Q(p to α ).( pSee), andFigure so they12.3.3. determine the polar α(p). See FigureWe 11.7. should keep in mind that the above construction must be correct becausewe know from Section 12.2 that such a hyperbolic polarity exists. It turns out that we could have chosenWe shouldany conic keep to in replace mind that G inthe the aboveabove constructionconstruction. must be correct because we know from Section 11.2 that such a hyperbolic polarity exists. It turns out that we could have chosen any conic to replace C in the above construction. One consequence of there being polarities is that statements about point conics can be dualized to statements about line conics. For example, Pascal’s Theorem becomes the statement of Briachon’s Theorem. The polar of a point conic C about C is the associated line conic for C. 6 6 CHAPTER 11. DUALITYCLASSICAL AND GEOMETRIES POLARITY MATH 4520 FALL 2017

Figure 12.3.3 Figure 11.7 One consequenceof there being polarities is that statements about point conics can be dualized to statements about line conics. For example, Pascal's Theorem becomesthe statement of Briachon's Theorem. The polar of a point conic C about C is the associated 11.4line conic Exercises for C .

1. Suppose that a point p in the Extended Euclidean plane has Affine coordinates Exercises: a 1. Suppose that a point p inthe Extendedp =Euclidean. plane has Affine coordinates b Describe the polar of Section 11.2p and= ~:) the polar of Section 11.3 of p in terms or the equation the line a(p) in Affine coordinates. Describe the polar of Section 12.2 and the polar of Section 12.3 of p in terms or 2. Considerthe theequation unit the circle line Ca(p)and in theAffine polarity coordinates.α of Section 11.3 of p in the Affine plane. 2. Consider the unit circle C and the polarity a of Section 12.3 of p in the Affine (a) Showplane that if p is not the center of C, then the line through the center of C and p a.is perpendicularShow that if p is to notα (thep). center of C, then the line thwough the center of C and (b) Supposep is perpendicular that the distanceto o.(p ). from the center of C to p is t. What is the distance b.from Suppose the center that the of distanceC to the from line theα (centerp)? of C to p is t. What is the distance from the center of C to the line o.(p)? 3. For3. theFor polarity the polarity of Problemof Problem 2 above,2 above, what what isis the image image of of a point a point circle, circle, other other than than C? C? 4. For4. theFor polarity the polarity of Problemof Problem 2 above,2 above, the the imageimage of the the following following point point ellipse ellipse is a is a line ellipse.line ellipse. (x/a)2 + (y/b)2 = 1. (x/a)2 + (y/b)2 = 1. What are the major and minor axes of this line ellipse? What5. areConsider the majorthe following and minormodel. axesAbstract of this points line are ellipse? the points inside the unit circle in the Euclidean plane as well as pajrs of antipodal points on the boundary. An 5. Consider the following model. Abstract points are the points inside the unit circle in the Euclidean plane as well as pairs of antipodal points on the boundary. An abstract line is defined as any line segment or circular arc between antipodal points on the boundary. See Figure 11.8. Show that this model satisfies the axioms for a projective plane.

6. Recall, in Chapter 8, our construction of the skew field H, the quaternions. Elements of H were defined as certain 2-by-2 matrices with entries in the complex field. We say that a one-to-one, onto function f : H → H is an anti-automorphism if for every x and y in H, f(x + y) = f(x) + f(y) DUALITY AND POLARITY 7

abstract line is defined as any line segmentor circular arc between antipodal points 11.4. EXERCISESon the boundary. SeeFigure 12.E.l. 7

Figure 12.E.l Figure 11.8 Show that this model satisfies the axioms for a projective plane. 6. Recall, in Chapter 8, our construction of the skew field H, the quaternions. El- andements of H were defined as certain 2 by 2 matrices with entries in the complex field. We say that a one-to-one, onto function f : H -+ H is an anti-automorphism f(xy) = f(y)f(x) if for every x and y in H, (a) Show that the function defined on which takes a matrix to its transpose is a f(x + y) = f(x)H + f(y) well-defined anti-automorphism of H. (b)and Suppose that f is any anti-automorphism of H such as in Part (a), for example. Consider the projective planef(xy) over= f(y)f(x)H using homogeneous coordinates. Show that that following function α is a well-defined polarity for this projective plane. a. Sho\\. that the function defined on H which takes a matrix to its transpose is a x \vell-defined anti-automorphism of H. b. Suppose that f is any anti-automorphismαy = f(x ),of f (Hy ),such f( zas) .in part a, for example. Consider the projective plane overz H using homogeneous coordinates. Show that that follo\...ing function a is a well-defined polarity for this projective plane.

Q (~) = [f(x), f(y ), j(z)] .