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 equiv- alent 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 :polefpoints of I. Note of πthatg ! the f linesexistence of πofg 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.