projective plane

projective plane

Projective planes

A

projective plane

is a plane (in various senses) where not only

•

for any two distinct

point

s,
there is exactly one

line

through both of them

(as usual, in things we call a “plane”), but also

•

for any two distinct

line

s,
there is exactly one

point

on both of them

(in other words, no

parallel

line

s). This gives

duality

between

point

s and

line

s: for any statement there is a corresponding statement
that swaps the words

point

and

line

, and swaps the phrases

lies on

and

passes through

(for which we can use the neutral

is

incident

with

).

A third axiom is commonly added to avoid degenerate cases:

•

there exist four points no three of which are

collinear

.

Both finite and

infinite

projective planes exist.

Here’s an example, just to show that such things exist: let

S

be the

unit
sphere

(the 2-dimensional

surface

x

2

+

y

2

+

z

2

=

1

embedded in

ℝ

3

). Call
every

great circle

(circle with radius 1 whose centre coincides with that of
the sphere) a

line

, and call every

pair of opposite points

on
the sphere a

point

. The notion of a

point

lying on a

line

is
well defined here, because each great circle (one of our

line

s) passing
through some point on the sphere also passes through its opposite point
(which together form one of our

point

s). Such a situation where different
flavors of “point” are discussed will arise again, and it is the reason
why the entities that act as

point

and

line

of a projective plane are typeset in their
own distinctive way in this entry.

And here’s a finite example, the

Fano plane

: the seven

point

s are labeled
with the

residue classes

(mod 7) and the seven

line

s are numbered likewise.
A

point

p

and a

line

q

are incident if and only if the

equation

x

2

=

q

-

p

does not
have a solution. Line

q

is incident with

point

s

q

+

1

,

q

+

2

,

q

+

4

and

point

p

is incident with

line

s

p

-

1

,

p

-

2

,

p

-

4

. Another way to get the
same plane (but with a quite different numbering): let

𝔽

2

be the

finite
field

of order 2 and

𝔽

2

3

the 3-dimensional

vector space

over

𝔽

.
The seven non-null vectors label our

point

s, and the

line

s are numbered
likewise. A

point

is incident with a

line

if and only if their

dot product

is zero
(for example:

point

(0,1,1) lies on

line

(1,1,1) because

0

⋅

1

+

1

⋅

1

+

1

⋅

1

=

2

=

0

).

Homogeneous coordinates

Historically, projective planes were formed by extending ordinary planes.

Let

Π

*

be a plane, a two-dimensional vector space over a field

𝔽

, where a point can be represented by its

Cartesian coordinates

(

X

,

Y

)

with

X

and

Y

∈

𝔽

(when

𝔽

=

ℝ

, this is the usual

Euclidean plane

). To specify a line

ℓ

, we can give a

linear equation

p

⁢

X

+

q

⁢

Y

+

r

=

0

(1)

satisfied by all the points on

ℓ

. The slope of the line is given by

-

p

/

q

(if

q

≠

0

) which is a single field

element

. Writing it as a

ratio

-

p

:

q

allows us to define slope for vertical lines as well,
where the ratio is

1

:

0

. This kind of

extension

of the field to include
some kind of

infinity

will be a recurrent theme.

There are many asymmetries between points and lines here. First of all, the
line is thought of as

being

a set of points (the points on the line)
while a point is not the same thing as the set of lines through that
point. This means we’re really interpreting equation (

1

) as

ℓ

=

def

{

(

X

,

Y

)

∣

p

⁢

X

+

q

⁢

Y

+

r

=

0

}

where

X

and

Y

are variables (

coordinates

of any point

∈

ℓ

) and

p

,

q

and

r

parameters specifying which line. We can redress this conceptual
imbalance by giving the line a kind of coordinates

[

p

,

q

,

r

]

. The linear
equation (

1

) now takes on a different

interpretation

: it is the
statement

that

the point

(

X

,

Y

)

and the line

[

p

,

q

,

r

]

are
incident, with both sets of coordinates on an equal footing.

Secondly, there are three coordinates for the line but only two for the point.
This is caused by the fact that line coordinate triples are not unique; only
the

ratio

of the coordinates matters. We can define

equivalence classes

[

p

:

q

:

r

]

=

def

{

[

f

p

,

f

q

,

f

r

]

∣

f

∈

𝔽

,

f

≠

0

}

of coordinate triples that

represent

the same line. The

p

,

q

and

r

used
to label a

[

p

:

q

:

r

]

are of course still just as non-unique (this
is in the same spirit as

labeling

a residue class or other coset by one of
its elements). One possible convention (for lines with

r

≠

0

at least)
is to choose the representative with

r

=

1

.

To find more

symmetry

between points and lines we could first define coordinate
triples for points with that same behaviour, so-called

homogeneous coordinates

(

x

,

y

,

z

)

=

def

(

x

/

z

,

y

/

z

)

which means

(

X

,

Y

,

1

)

and more generally

(

f

⁢

X

,

f

⁢

Y

,

f

)

for

f

≠

0

are new
names for the point

(

X

,

Y

)

, and then define the equivalence classes

(

x

:

y

:

z

)

=

def

{

(

f

x

,

f

y

,

f

z

)

∣

f

∈

𝔽

,

f

≠

0

}

to formally put all those different names for the same point back into a
single box. This exercise gives the statement

p

⁢

x

+

q

⁢

y

+

r

⁢

z

=

0

(2)

that

(

x

:

y

:

z

)

and

[

p

:

q

:

r

]

are incident a pleasing symmetry.

The line and points at infinity

Thus far, we cannot have any point

(

x

:

y

:

z

)

where

z

=

0

(it does not
correspond to any point

(

X

,

Y

)

in the plane). By contrast,

[

p

:

q

:

r

]

can have

r

=

0

(for a line through the arbitrary origin of the

X

⁢

Y

coordinate
frame). For lines only the case

p

=

q

=

0

is missing, whereas

x

=

y

=

0

is
fine for a point. Thus, by trying to make points and lines as

similar

as
possible we have unearthed their essential

difference

algebraically.

The geometric difference is that lines can be parallel, and it is easy to
see this is the same difference. For any two points

(

x

,

y

,

z

)

and

(

x

′

,

y

′

,

z

′

)

we can find a line

[

y

⁢

z

′

-

z

⁢

y

′

,

z

⁢

x

′

-

x

⁢

z

′

,

x

⁢

y

′

-

y

⁢

x

′

]

that passes through both, and it is a valid line (first two coordinates not
both zero) if the points are valid (

z

and

z

′

nonzero) and distinct,
but attempting the dual construction reveals pairs of valid lines that do
not

intersect

in a valid point.

We now extend the plane

Π

*

to a new kind of plane

Π

,
inheriting all the points and lines of

Π

*

as

point

s and

line

s of

Π

and co-opting additionally

•

the new

line

[

0

:

0

:

1

]

— note only one new

line

is
needed as all triples

[

0

,

0

,

f

]

(with

f

≠

0

)
fall in this equivalence class, and

•

the new

point

s

(

1

:

f

:

0

)

(comprising all

(

x

,

y

,

0

)

with nonzero

x

,
using

f

=

y

/

x

), as well as

(

0

:

1

:

0

)

(this class has those

(

x

,

y

,

0

)

where

x

=

0

).

The only ratio excluded for a

point

is

(

0

:

0

:

0

)

and for a

line

[

0

:

0

:

0

]

,
making the situation

symmetric

. Note all the new

point

s “lie on” the new

line

and none of the old ones do, which can be seen by applying (

2

).
Any of the old

line

s acquires one of the new

point

s,

line

s that were parallel
get the same new

point

and

line

s that weren’t get different new

point

s. This
prevents any pair of

line

s already intersecting in an old

point

intersecting in
a new one as well, and provides any pair that didn’t yet intersect a “place”
where to do so.

The new

point

s are what is shared by

parallel lines

, so correspond to

directions

(pairs of opposite directions, in fact). They are
called

points at infinity

and the new

line

comprising them the

line at infinity

.

The embedding in

𝔽

3

The extension of

Π

*

to

Π

in the previous

section

has an
immediate geometric interpretation. Interpret every

(

x

,

y

,

z

)

as a distinct
point in

𝔽

3

∖

(

0

,

0

,

0

)

. The equivalence classes

(

x

:

y

:

z

)

, or
strictly speaking

(

x

:

y

:

z

)

∪

{

(

0

,

0

,

0

)

}

, are now lines through the origin:

•

the

point

s of

Π

can be regarded as 1-dimensional

subspaces

of

𝔽

3

.

We could equate the

[

p

:

q

:

r

]

with lines through the origin as well, but it
will turn out to be more convenient to identify them with the planes through
the origin

⟂

those lines. Those planes consist of precisely those vectors

(

x

,

y

,

z

)

∈

𝔽

3

that are

⟂

the vector

(

p

,

q

,

r

)

∈

𝔽

3

.
In this way

•

the

line

s of

Π

can be regarded as 2-dimensional subspaces of

𝔽

3

.

The reason this is more convenient is that now, by construction, all the
points (vectors) in the 1-d subspace corresponding to a

point

(

x

,

y

,

z

)

are

⟂

the vector

(

p

,

q

,

r

)

if and only if equation (

2

) holds. In other
words, those points lie inside the 2-d subspace corresponding to the

line

[

p

:

q

:

r

]

if and only if the

point

is incident with this

line

:

•

point

P is incident with

line

ℓ

if and only if the line corresponding
to P

⊂

the plane corresponding to

ℓ

.

We can collapse back from 3

dimensions

to 2 dimensions in various ways.

First, intersect the 1- and 2-dimensional subspaces representing

point

s and

line

s with the plane in

𝔽

3

with

z

=

1

; call this plane

Π

1

*

.

∘

The

point

s (lines)

(

x

:

y

:

z

)

with

z

≠

0

each contain one
point

(

x

/

z

,

y

/

z

,

1

)

in

Π

1

*

. The

point

s
with

z

=

0

(lines in

𝔽

3

parallel to

Π

1

*

)
don’t have any point in

Π

1

*

.

∘

The

line

s (planes)

[

p

:

q

:

r

]

with

p

and

q

not both zero
intersect

Π

1

*

in a line with equation (

1

).
The only missing

line

(plane) is the one through the origin
parallel to

Π

1

*

, which contains all the missing

point

s.

So the plane

Π

1

*

is exactly the plane

Π

*

we started off
with, with an extra third coordinate 1 tacked on at the end.

Alternatively, intersect the 1- and 2-dimensional subspaces representing

point

s and

line

s with the sphere

x

2

+

y

2

+

z

2

=

1

.

∘

The

intersection

of a

point

(line through the origin) with this
sphere is two opposite points.

∘

The intersection of a

line

(plane through the origin) with the
sphere is a great circle.

When

𝔽

=

ℝ

, this is exactly the first example in this article.

The second example there (the finite one, the Fano plane) can be seen as
the

embedding

in

𝔽

3

when

𝔽

=

𝔽

2

. Its other

representation

showed it also has a cyclic symmetry modulo 7 (many more than one, in fact
— its

automorphism group

has order 168).

A health warning is in order: the coordinatisation here gives the
spurious idea there is some special

relation

between the

point

(

a

:

b

:

c

)

and the

line

[

a

:

b

:

c

]

. In the sphere in

ℝ

3

for example, such
a

point

plays the rôle of poles relative to the

line

as equator. The whole
point of

projective geometry

however, which we will not pursue further in
this entry, is that only incidence between

point

s and

line

s is considered
meaningful, and metric considerations (

distances

and angles) are ignored.
If we redo the

ℝ

3

example with any arbitrary set of three

independent

vectors as basis, we get projective planes all

isomorphic

with each other as far as
incidence is concerned, but the pairs

(

a

:

b

:

c

)

and

[

a

:

b

:

c

]

that
end up with the “same” coordinates are different in each version.

In the finite example too, if we choose a basis different from (1,0,0),
(0,1,0), (0,0,1) we can find the same plane in many different guises:
there are 7 ways to choose the first basis vector, 6 ways to find a
second distinct from the first, and 4 ways for a third not in the plane
of the first two, and

7

⋅

6

⋅

4

=

168

.

Classical finite planes

The constructions above can be carried out for

𝔽

2

=

ℝ

2

where they
extend the usual Euclidean plane to “the” projective plane, but equally
well for any other field, such as

finite fields

(aka Galois fields).
Such fields have

q

=

p

d

elements for

p

any prime and

d

any

positive
integer

.

The

Π

*

that takes the place of the Euclidean plane is now called a
(finite)

affine plane

. For

d

=

1

i.e.

q

=

p

it is still fairly easy
to visualise. Now the field is

arithmetic

(mod

p

) and the affine plane
is a grid of

q

×

q

points, with lines connecting them at all possible

integer

ratio slopes (you need to wrap top to bottom and left to right,
in modulo fashion, if you attempt to draw it). There are

q

slopes

f

:

1

(in a field, every nonzero element has a

multiplicative inverse

so every

a

:

b

is some

a

⁢

b

-

1

:

1

), and one slope

1

:

0

for vertical lines. For each of the

q

+

1

slopes there are

q

parallel lines, making

q

2

+

q

lines in all.

The (finite)

projective plane

Π

adds

q

+

1

point

s at infinity
(one for each slope, each direction shared by a bunch of parallel lines)
to the

q

2

points of the grid,

q

2

+

q

+

1

point

s in all. And it adds one

line

at infinity to the

q

2

+

q

lines we had, making

q

2

+

q

+

1

line

s as well.
So

•

the plane has

q

2

+

q

+

1

point

s and

q

2

+

q

+

1

line

s.

This is also evident from the embedding in

𝔽

3

∖

{

(

0

,

0

,

0

)

}

where each time

q

-

1

non-(0,0,0) points lie on the same

point

(line through
the origin), and

(

q

3

-

1

)

/

(

q

-

1

)

=

q

2

+

q

+

1

. Each

line

(plane through the
origin) is

⟂

such a line through the origin, so the numbers are the
same. We also have that

•

every

line

is incident with

q

+

1

point

s; every

point

with

q

+

1

line

s

because each

line

(plane through the origin) contains

q

2

-

1

non-(0,0,0)
points, and

(

q

2

-

1

)

/

(

q

-

1

)

=

q

+

1

. The simplest way to see the number the
other way round is also

q

+

1

is calling

point

s

line

s and vice versa and
embedding the thing in another

𝔽

3

; the incidence relation
(

2

) is unchanged under this swap.

The field-based planes constructed here are the

classical

planes,
also called

Desarguesian

because Desargues’s

theorem

holds in them,
and

Pappian

because Pappus’s theorem holds.

Π

⁢

α

´

⁢

π

⁢

π

⁢

o

⁢

ς

(Pappus) was a 4th century Alexandrine
mathematician and Desargues a 17th century French one.

It can be shown that Pappus’s theorem holds in precisely those planes
constructed in this way with

𝔽

a field (

commutative

division ring

),
whereas Desargues’s theorem holds whenever

𝔽

is a skew field (division
ring). For finite planes both conditions are the same by Wedderburn’s theorem,
so Desarguesian and Pappian are synonyms. For infinite planes you can have
Desarguesian non-Pappian ones, for

instance

if

𝔽

is taken to be the

quaternions

.

The question arises what other

algebraic structures

other than
(skew) fields can produce projective planes. See also the

http://planetmath.org/node/6943

finite projective planes

entry

.

Projective spaces

The same construction with homogeneous coordinates can be carried out in different
numbers of dimensions. This extends

𝔽

1

to the

projective line

with
for a finite field

q

+

1

elements, labeled by

𝔽

∪

{

∞

}

where

∞

is any item not in

𝔽

. It likewise extends

𝔽

n

for

n

>

2

to higher

projective spaces

with

(

q

n

+

1

-

1

)

/

(

q

-

1

)

elements.

For

n

≠

2

however these classical constructions give the only possible
projective spaces; there is nothing corresponding to the wild

variety

of
non-classical projective planes we find for

n

=

2

.

Title

projective plane

Canonical name

ProjectivePlane

Date of creation

2013-03-22 15:11:05

Last modified on

2013-03-22 15:11:05

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

13

Author

yark (2760)

Entry type

Topic

Classification

msc 51E15

Classification

msc 51N15

Classification

msc 05B25

Related topic

FiniteProjectivePlane4

Related topic

IncidenceStructures

Related topic

Geometry

Related topic

LinearSpace2

Related topic

AxiomaticProjectiveGeometry

Related topic

LinearSpace3

Generated on Fri Feb 9 22:00:27 2018 by

LaTeXML