Chapter 11
Duality and Polarity
Math 4520, Spring 2015
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.
2
CHAPTER 11. DUALITY AND POLARITY
MATH 4520, SPRING 2015
In order to get a better understanding of more of the geometry than the incidence structure, 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 Homogeneous models 1 and 2. We say that two points p and q on the unit sphere in real 3-space are
antipodal
if q = −p. So two distinct
pointsGEOMETRIES
on the unit sphere are antipodal if and only if
CLASSICAL
2
they lie on a line through the origin. We say that a great circle is the intersection of a plane
and only
they lie
on athe
lineunit
through
the origin. We say that a great circle is the intersection
through
the iforigin
with
sphere.
of
a
plane
through
the
origin
with
the
unitthe
sphere.
We now define the Sphere Model 1 for
real projective plane. Here a projective point
We now define the SphereModel 1 for the real projective plane. Here a projective point
is a ispair
of antipodal
unitsphere.
sphere.A A
projective
is a great
on
a pair
of antipodalpoints
pointson
onthe
the real
real unit
projective
line line
is a great
circle circle
on
the unit
sphere.
Incidence
is
just
set
containment.
Clearly
this
model
is
equivalent
to
the
the unit sphere. Incidence is just set containment. Clearly this model is equivalent to the
Homogeneous
1 over
the
field.
Figure
11.1
CLASSICAL
GEOMETRIES
2 Model
Homogeneous
Model1
over
thereal
real
field. See
See
Figure 12.1.1
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 SphereModel 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
Figure
12.1.1 However, a projective line is defined to
point is defined to be a pair of antipodal
points.
be a pair
antipodal
points.
Incidence
isfor
perpendicularity.
Finally
weofdefine
the
Sphere
Model
the
projective
plane.
Again a projective
Finally
v.'e
define
the Sphere
Model22
for
the
realreal
projective
plane. Again
a projective
point
is
defined
to
be
a
pair
of
antipodal
points.
However,
a
projective
line
is
defined
We
summarize
this
in
the
following
table.
point is defined to be a pair of antipodal points. However, a projective linetois defined to be
be a pair of antipodal points. Incidence is perpendicularity.
a pair of antipodal
points. this
Incidence
is perpendicularity.
We summarize
in the following
table.
We summarize this in Figure 11.2.
The Plane
Points
The Plane
Extended
Euclidean
Points
Extended
Plane
Euclidean
Plane
Finite
points
Finite points
& ideal
points
& ideal
points at
at
infinity
infinity
Homogeneous
Homogeneous
coordinates
coordinates
1
1
Homogeneous
Homogeneous
coordinates
coordinates
2
2
Sphere model 1
Sphere model 1
Sphere model 2
Lines through
Lines
through
O
O
Ordinary points
at "is
Ordinary points
&&thethe
lineline
at
member
of"
of"
infinity
infinity
O
except
O
Antipodal
Greatexcept
circles on0
the sphere
points on the
"is a
a member
Set containment
Planes through
Set containment
Planes through
O
Lines through O
except
0
Lines
through
sphere
Sphere model 2
Incidence
Lines through O
except O O
Lines through
Antipodal
sphere
pointsAntipodal
on the
points
on the
sphere
Incidence
Lines
Lines
Perpendicularity
O
Set contajnment
Great circles on
the sphere
Antipodal
points on the
sphere
Antipodal
Antipodal
points on the
points on the
sphere Figure 11.2 sphere
Perpendicularity
Set contajnment
Perpendicularity
Perpendicularity
It is a good exercise to convince yourself that all of these models for the real projective
plane are equivalent.
11.2. POLARITIES
11.2
DUALITY
AND POLARITY
3
3
Polarities
It is a good exerciseto 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.
Polarities
In fact, the12.2
word
“point” and the word “line” can be interchanged and the axioms remain
Recall
that our axioms
for a projective
plane had
a symmetry
with words
respect points
the same. Any
statement
in projective
geometry
can
reverse the
“line”and
and “point”
lines. In fact, the word "point" and the word "line" can be interchanged and the axioms
and it should
make
sense.Any
But
will one
statement
be true
other"line"
is? Ifand
the truth of
remain
the same.
statement
in projective
geometry
canwhen
reversethe
the words
one statement
directly
from
theButoriginal
axioms,
answer
will
"point"follows
and it should
make
sense.
will one three
statement
be true the
when
the other
is? be
If yes. But
truth of one statement
directly
from the original
three
axioms, therelied
answeron
willusing more
some of thethestatements
that we follows
proved,
for example
Pascal’s
Theorem,
be yes. But some of the statements that we proved, for example Pascal's Theorem, relied
than our three
axioms.
So our
we three
look axioms.
for a more
relating
therelating
pointstheand lines.
on using
more than
So weexplicit
look for away
moreof
explicit
way of
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
between
projective
planes was an incidence
{points
of 7!"} -+
{lines of 7!"
}
preserving function taking points to points and lines to lines. Here we define a similar notion
be a one-to-one onto function suchthat for any point p of 7!' and any line 1 of 7!', p and 1
but with points
and lines being interchanged. Let π be any projective plane and let
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) the
of I. Note
that→
the{lines
existence
of such a polarity says that,
α :pole
{points
of π}
of π}
in a certain sense, points and lines are interchangeable.
We present a few examples. Let us take the Homogeneous Coordinates Model I. We can
take theonto
polarfunction
of a line through
the origin
to be point
the plane
the any
originline
perpendicular
be a one-to-one
such that
for any
p through
of π and
l of π, p and l are
to that line, and so the pole of plane is the line through the origin perpendicular to it.
incident if and only if α(p) and α(l) are incident, where we use α to denote 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,
since
domain
and circle
range
of pair
α are
different.
Wethat
think
to them,
andthe
the pole
of a great
is the
of antipodal
points
are of α as a
on the between
line throughthe
the set
origin
the set
planeofoflines.)
the greatSuch
circle. aSofunction
the pole ofis called a
correspondence
of perpendicular
points andtothe
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 equidistant 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, SPRING 2015
p
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
0 = αy y  = x, y, −z y  = x2 + y 2 − z 2 .
z
z
z
So a point is incident to its polar if and only if it lies on a conic, the circle x2 + y 2 = 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
11.3. MORE POLARITIES
DUALITY
AND POLARITY
AND POLARITY
5
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)
11.5is 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
defined
above.
contradicts 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.
0
is 0a) point
inside the
Choose
two α(p
distinct
II and
12that
Figure
12.3.2
and since Suppose
p0 is onthat
C, P
α(p
is incident
tocircle
p0 asG.
well.
Thus
) =lines
α(p),
which
contradicts
are incident to p. Reversethe construction of Figure 12.3.2 to find the poles PI = Q(11)
that α isand
one-to-one.
Thus α(p)
isP2
tangent
to C.to Q(p), and so they determine the polar
P2that
= 0.(12).
PI and
are
Suppose
P is Then
a point
outside
theincident
circle G.
Let TI and T2 be the two tangents to G
Suppose
p they
is a point outside
theQ(p
circle
C.
Let 12.3.3.
T and T2 be the two tangents to C
Q(p ), that
and so
the polar
). See
Figure
that are
incident
to p.determine
See"Figure
12.3.2.
So Q(T1)
and 1Q(T2) are
the points of tangency
We
should
keep
in
mind
that
the
above
construction
must
be correct
becauseof
we know
that
incident
to p. See Figure
11.6.
So α(Tand
α(Tare
the
points
of T1
1 ) and
2 ) are
of are
TI from
and
T2,
respectively,
on
G,
and
Q(T1)
Q(T2)
both
incident
to tangency
Q(p).
Section 12.2 that such a hyperbolic polarity exists. It turns out that we could
have Thus
andQ(p)
T2 , is
respectively,
on C, andbyα(T
)two
andpoints
α(T2 )ofare
both incident
to T2.
α(p). Thus α(p) is the
the line
determined
tangency
of TI and
chosen
any conic
to replace Gthe
in 1the
above construction.
line determined
by Pthe
points
of tangency
1 and Ttwo
2.
Supposethat
is two
a point
inside
the circle of
G.TChoose
distinct lines II and 12that
are
incident
to
p.
Reverse
the
construction
of
Figure
12.3.2
to
find the lines
poleslPI
= Q(11)
Suppose that p is a point inside the circle C. Choose two distinct
1 and l2 that
P2 = to
0.(12).
Then PIthe
andconstruction
P2 are incident
to Q(p11.6
), and
they
polar
areand
incident
p. Reverse
of Figure
to so
find
thedetermine
poles p1 the
= α(l
1 ) and
Q(p
),
and
so
they
determine
the
polar
Q(p
).
See
Figure
12.3.3.
p2 = α(l2 ). Then p1 and p2 are incident to α(p), and so they determine the polar α(p). See
should keep in mind that the above construction must be correct becausewe know
FigureWe
11.7.
from Section 12.2 that such a hyperbolic polarity exists. It turns out that we could have
We should
keep to
in replace
mind that
chosen
any conic
G inthe
theabove
aboveconstruction
construction. 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
CLASSICAL GEOMETRIES
CHAPTER 11. DUALITY
AND POLARITY
6
Figure
MATH 4520, SPRING 2015
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
line conic for C .
11.4
Exercises
1. Suppose that a point p in the Extended Euclidean plane has Affine coordinates
Exercises:
a
1. Suppose that a point p inthe Extended
p =Euclidean
. plane has Affine coordinates
b
= ~:)the polar of Section 11.3 of p in terms or the
Describe the polar of Section 11.2p and
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. Consider
unit the
circle
andin the
polarity
α of Section 11.3 of p in the Affine plane.
the the
equation
line Ca(p)
Affine
coordinates.
2.
Consider the unit circle C and the polarity a of Section 12.3 of p in the Affine
(a) Show
plane that if p is not the center of C, then the line through the center of C and p
is
a. perpendicular
Show that if p isto
notα(p).
the center of C, then the line thwough the center of C and
p is perpendicular
to o.(p ). from the center of C to p is t. What is the distance
(b) Suppose
that the distance
b. Suppose that the distance from the center of C to p is t. What is the distance
from
the center of C to the line α(p)?
from the center of C to the line o.(p)?
the polarity
of Problem
2 above,what
what is
is the
the image
circle,
otherother
than than C?
3. For3.theFor
polarity
of Problem
2 above,
imageofofa point
a point
circle,
C?
the polarity
of Problem
2 above,the
theimage
image of the
ellipse
is a is a line
4. For4.theForpolarity
of Problem
2 above,
thefollowing
followingpoint
point
ellipse
line
ellipse.
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?
What
are the major and minor axes of this line ellipse?
5. Consider the following model. Abstract points are 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
7
AND POLARITY
abstract line is defined as any line segmentor circular arc between antipodal points
11.4. EXERCISES
7
on the boundary. SeeFigure 12.E.l.
Figure
12.E.l
Figure 11.8
Show that this model satisfies the axioms for a projective plane.
Recall,
in Chapter 8, our construction of the skew field H, the quaternions. El6.
ements of H were defined as certain 2 by 2 matrices with entries in the complex
and
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 H which takes a matrix to its transpose is a
f(x + y) = f(x) + f(y)
well-defined anti-automorphism of H.
(b)
andSuppose that f is any anti-automorphism of H such as in Part (a), for example.
Consider the projective plane
H using homogeneous coordinates. Show that
f(xy) over
= f(y)f(x)
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.
f (z)
αy  = f (x), off (y),
b. Suppose that f is any anti-automorphism
H such
as .in part a, for example.
z
Consider the projective plane over
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)]
.