The
Duality
Principle
in Geometry
Date: April 2006
Chad Christianson
Mathematics Discipline
University of Minnesota, Morris
Morris, MN 56267,
chri1008@morris.umn.edu
Faculty Advisor:
Michael O’Reil
l
y
Mathematics Discipline
University of Minnesota, Morris
Morris, MN 56267,
oreillym@morris.umn.edu
Abstract:
The Duality Principle
is difficult to define in mathematics since depending
on which field y
ou are dealing with the definition is slightly changed. In
general,
duality
is simply quality of being two fold.
This paper will discuss
the duality principle in projective geometry through the use of an axiomatic
approach. This paper will also discuss
how duality is used in projective
geometry
and how it is viewed from a classical aspect.
Christianson, Page
2
1.
Introduction to Duality
Duality
is a very important topic in mathematics. It is most commonly seen
in projective geometry, but is also seen in many
other areas. As previously
stated duality is the quality of being two fold, which in mathematics means
given one conclusion we can easily reach another conclusion which is
equally important.
There have been many important theorems throughout the years t
hat have
been proven and then later shown that they have a dual which is just as
useful.
Definition:
In a geometry aspect, we say that two theorems are dual to each
other if we can take one theorem and replace terms such as point for line and
line for p
oint (in two dimensions) and receive a second theorem that is
equally important.
The dual of a theorem is one example of duality in
mathematics
.
There are many examples of dual theorems such as Menelaus’
Theorem and Ceva’s Theorem, which we will discuss in
our next section
.
[PlanetMath] [Coxeter]
2.
Classical Duality
In this section we will look at duality in a classical geometry aspect. We
will do this by looking at the Menelaus’ and Ceva’s Theorems and also the
Pascal’s and Brianchon’s Theorems.
2.1 Mene
laus’
and Ceva’s Theorems
The most common example of dual theorems in a classical sense is the
Menelaus’ Theorem and Ceva’s Theorems.
Definition:
Define
[xy]
as
the distance between points x and y. We will
define a ratio of distances as either positiv
e or negative as follows; given
Christianson, Page
3
three points A, B, and C,
is
positive
if B lies between A and C,
is
negative
if B lies on the extension of the line segment AC.
The Menelaus’ Theorem states the following; giv
en a triangle ABC
t
hen the
points a, b, c located on the (extended) edges BC, AC, AB of the triangle
ABC is collinear if and only if
(See Figure 1)
Figure 1
[PlanetMath]
Definition:
A
cevian
is a line segment joining a vertex of a triangle to any
given point on the opposite side of the triangle (Figure 2).
The dual of Menelaus’ Theorem is Ceva’s Theorem which states; giv
en a
triangle ABC, if three cevians
AX, BY, and CZ are concurrent then
.
Figure 2
[PlanetMath]
Christianson, Page
4
Both of these theorems ca
n be proven individually or given one of the
theorems we can prove the other.
Let’s
look at how we can get Ceva’s Theorem using only Menelaus’
Theorem and our basic understandings of geometry. First
let’s
take our
triangle ABC from Figure 2.
1. Given
BAX:
is a transversal:
2.
(Menelaus’ Theorem)
3.
Given
ACX:
is a transversal:
4.
(Menelaus’ Theorem)
5. Multiplying (2
) x (4);
This shows how given Menelaus’ Theorem we can achieve Ceva’s theorem.
I believe this is also because cross

ratios are preserved by the duality
principle but I did not look deeper into this matter.
This relation be
tween the
two theorems is what makes them duals of each other
.
[PlanetMath]
2.2 Pascal’s and Brianchon’s Theorems
Much of the early work in duality was done in conic geometry. When
Desargues started working with projective geometry he was really intere
sted
in the ideas of conics.
At the time most mathematicians felt that all of the
work that could be done with conics had already been achieved, but when
Desargues came out with his new work many mathematicians were
pleasantly surprised with the new ideas
that he had discovered.
Two dual theorems which are examples of classical duality that also deal
with conics are Pascal’s Theorem and Brianchon’s Theorem. Pascal’s
Christianson, Page
5
Theorem states that, given a hexagon inscribed in a conic section, the three
points at wh
ich the pairs of the opposite sides meet, lie on a straight line.
This straight line is called the
Pascal Line
(Figure 3)
.
Figure 3
[Kline]
The dual of Pascal’s Theorem is Brianchon’s Theorem. Brianchon’s
Theorem states that,
given a hexagon circumsc
ribed on a conic section, the
lines joining opposite polygon vertices meet in a single point (Figure 4).
Figure 4
[Kline]
The easiest way to see that these two theorems are actually duals of
each other is to write the two side by side.
Pascal’s Theorem
Brianchon’s
Theorem (
Dual of
Pascal)
If we take six points, A, B, C, D, E, and
If we take six lines
a, b, c, d, e,
and
F on the point conic, then the lines
f
on the line conic, then the points
which join A and B and D and E
which join
a
and
b
and
d
and
e
are
meet in a point P; the lines
joined by the line
p
; the points
which join B and C and E and F
which join
b
and
c
and
e
and
f
are
Christianson, Page
6
meet in a point Q;
joined by the line
q
;
the lines which join C and D and F and
the points which join
c
and
d
and
f
A meet in a point R.
and
a
are joined by the line
r
.
The three points P, Q, and R lie on
The three lines
p, q,
and
r
are on
one line
l
.
one point L.
In this example the words points have been replaced by lines and a few
minor words have been cha
nged in order to make the language clear. The
two

dimensional principle of duality states that every definition remains
significant and every theorem remains true when we interchange the words
points and lines. This also holds for three

dimensional space
when points,
lines, and planes are interchanged with planes, lines, and points. In three

dimensional space we say that lines are self

dual since we replace lines with
lines. In two

dimensions we can take the dual of a triangle which consists of
its verti
ces and sides is again a triangle consisting of its sides and vertices.
Therefore, a triangle is a self

dual figure
.
[Kline]
3
. Projective Geometry
In this section we will look at projective geometry and use some of its
axioms to prove
Desargues’s Theore
m and its dual
in projective geometry.
Projective Geometry is the study of geometry where properties remain the
same under transformation.
3
.1 Introduction to Projective Geometry
Another area of mathematics that we should look into is projective
geometr
y. Plane (Euclidian) geometry is the geometry of lines and circles,
which can be created using only a straight

edge and a compass. Projective
geometry was created with the use of only the straight

edge. Though this
geometry deals with points, lines and
planes there is no attempt to measure
distances between two points or the angle between two lines. It also brings
up the possibility that two lines that are parallel meet at a point at infinity.
This can be easily seen by standing on a railroad tracks.
The two rails are
definitely parallel to each other but still appear to meet at some point on the
Christianson, Page
7
horizon, also known as a vanishing point. We can also see this in drawings
of tiles on a vertical canvas. The square tiles no longer appear to be square
due
to the alteration of the angles and lengths of the sides.
In projective
geometry
curves are simply considered conics.
Joseph Gergonne noted the principle of duality characterizing projective
plane geometry: given any theorem or definition in projecti
ve geometry,
substituting
point
for
line
,
lie on
for
pass through
,
collinear
for
concurrent
,
intersection
for
join
, or vice versa, results in another theorem or definition,
the dual of the first. This means that in projective geometry all theorems and
def
initions occur in “dual pairs”.
Here we will list
seven
basic axioms in projective geometry.
1. There exist a point and a line that
are
not incident.
2. Every line is incident with at least three distinct points
3. Any two distinct points are incident wit
h just one line.
4. If A, B, C, D, are four distinct points such that AB meets CD, then AC
meets BD.
(Figure 7
)
5. If ABC is a plane, there is at least one point not in the plane ABC.
6. Any two distinct planes have at least two common points.
Definition:
An
elementary correspondence
is a projection that takes a
point X to a line x through some point of reference. (See Figure 5)
Definition:
A
projectivity
is the product of two elementary
correspondences.
(See Figure 6)
7. If a projectivity leaves invariant
each of three distinct points on a
line, it leaves invariant every point on the line.
Christianson, Page
8
Figure 5
Figure 6
Figure
7
[Coxeter]
Christianson, Page
9
All infinite geometries can b
e derived as
special
cases of projective
geometry by simply adding the extra notions and axioms.
3
.2 Proof of Desargues’s Theorem and its Dual
Definition:
We say that two figures are
perspective from a point O
if pairs
of corresponding points are joine
d by lines through O, and that two figures
are
perspective from a line o
if pairs of corresponding lines meet on o.
Desargues’s Theorem states that;
If two triangles are perspective from a
point they are perspective from a line
.
The Dual of Desargues’s
Theorem states;
If two triangles are perspective
from a line they are perspective from a point
.
Let us first prove the dual of Desargues’s Theorem,
If two triangles are
perspective from a line they are perspective from a point
.
Proof. Let two triangles,
PQR and P’Q’R’, be perspective from a line
o.
This means that
o
contains three points D, E, F, such that D lies on both QR
and Q’R’, E on both RP and R’P’, F on both PQ and P’Q’. We need to
prove that the three lines PP’, QQ’, RR’ all pass through one po
int
O
, see
figure 8
. We will set up two possible cases depending on if the two given
triangles are in distinct planes or lie in the same plane.
Christianson, Page
10
Figure 8
[Coxeter]
Case 1: Axiom 4 says since QR meets Q’R’, QQ’ meets RR’. Similarly
RR’ meets PP’, and P
P’ meets QQ’. Therefore the three lines PP’, QQ’,
RR’ all meet one anther. If the planes PQR and P’Q’R’ are distinct then the
three lines must be concurrent; otherwise they would form a triangle which
would lie in both planes.
Case 2: If PQR and P’Q’R’
are in one plane we can draw in a different
plane through
o
, three nonconcurrent lines through D, E, F, to form a
triangle
with
through D,
through E, and
through F.
This new triangle is perspective from
o
with both PQR and P’Q’R’. Since
these are noncoplanar triangles, the three lines P
, Q
, R
all pass
through one point S, and the three lines P’
, Q’
, R’
all pass through
another point S’, where S and S’ are distinct p
oints. Since
lies on both PS
and P’S’, axiom 4 states that SS’ meets PP’. Similarly SS’ meets both QQ’
and RR’. Therefore the three lines PP’, QQ’, RR’ all pass through the point
O.
Now for the proof of Desargues’s Theorem,
If tw
o triangles are perspective
from a point they are perspective from a line
.
Proof: Let two triangles PQR and P’Q’R’ be perspective from a point
O
.
Axiom 4 states that the three pairs of corresponding sides meet, say at D, E,
F respectively. Must show tha
t these three points are collinear
, see Figure 8
.
Look at triangles PP’E and QQ’D. Since the pairs of the corresponding sides
Christianson, Page
11
meet in the three collinear points R’, R, O, these triangles are perspective
from a line, and therefore, by the first proof, pers
pective from a point, call it
F. Therefore the three points E, D, F are collinear.
[Coxeter]
4
. Conclusion
and Possible Areas of
Expansion
This paper shows that duality is a very important and useful topic in
mathematics.
One way this could be expande
d is to look at other areas of
mathematics to see how the duality principle is treated there
.
Another area
that could be looked into is how and why cross

ratios are preserved.
There
are many different duality theorems out there that could be
analyzed
or
possibly even expanded upon. Also the areas of differential geometry is a
very interesting and deep topic where this project could be expanded into.
5
. Acknowledgements
I would like to thank Professor
Michael O’Reilly
for his help advising me
during thi
s project. I would like to thank Professor Peh Ng for being my
second reader for this paper.
Christianson, Page
12
6
. References
[1]
Birkhoff, Garrett; Mac Lane, Saunders.
A Survey of Modern Algebra.
The Macmillan Company, 1965.
[2]
Coxeter, H.S.M.
Projective G
eometry
. University of Toronto Press,
1974.
[3]
Curtis, Charles W.
Linear Algebra, An Introductory Approach.
Allyn
and Bacon Inc., 1963.
[4]
“Duality Principle.” From
PlanetMath

An Encyclopedia Web
Resource.
http://planetmath.org/encyclopedia/DualityPrinciple.html
[5]
Kline,
Morris.
Mathematical Thought From Ancient to Modern Times.
Oxford University Press, 1972.
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