axiomatic projective geometry

axiomatic projective geometry

The Language and Axioms

The

language

of

axiomatic projective geometry

is a

first-order language

consisting of two sorts: point and line, and one binary

predicate

I

called “incidence” between points and lines. There are various ways to call

I

⁢

(

p

,

L

)

: point

p

is

incident

with line

L

,

L

is incident with

p

,

p

lies on

L

,

p

is on

L

, or

L

passes through

p

.

There are three axioms for

projective geometry

:

Axiom 1

. For any two distinct points, there is exactly one line incident with the points.

If

p

,

q

are points, we write

p

⁢

q

or

q

⁢

p

for the line incident with

p

and

q

.

In order to state the next axiom, we define the

concept

of meet or

intersection

: two lines meet or intersect if there is a point incident with both lines, and that they are

parallel

if no such a point exists.

Axiom 2

. (Veblen’s axiom) For any four pairwise distinct points

p

,

q

,

r

,

s

, if

p

⁢

q

meets

r

⁢

s

, then

p

⁢

r

meets

q

⁢

s

.

A set of points is said to be

collinear

if there is a line incident with each of the points. Axiom 2 can be equivalently stated as: if points

x

,

q

,

s

are not collinear, and if

p

≠

x

is a point on

x

⁢

q

, and

r

≠

x

is a point on

x

⁢

s

, then

p

⁢

r

meets

q

⁢

s

.

Axiom 3

. Each line is incident with at least three points.

Models

A model of projective geometry is called a projective space. It consists of a set

𝒫

of points, a set

ℒ

of lines, and a

relation

I

⊆

𝒫

×

ℒ

such that the three axioms above are satisfied. In other words, a projective space is an incidence structure satisfying the conditions stated by the axioms. Here, lines and blocks are synonymous. In

addition

, if two lines are incident with the identical set of points, this set must contain at least three points, and so by Axiom 1, the lines are identical. In other words, any projective space is a simple incidence structure, and therefore any line may be identified with the set of points on it, which we will do for the remainder of the entry.

∅

, a single point with no lines, a single line with at least three points, are all projective spaces. A prototypical example of a projective space is

P

⁢

G

⁢

(

V

)

, where

V

is a

vector space

over some division ring

k

. The points and lines of

P

⁢

G

⁢

(

V

)

are one- and two-dimensional

subspaces

and of

V

, respectively, and incidence is

set inclusion

.

Axiom 3 is sometimes known as the non-degeneracy axiom. With it, one can show that any two lines are equipolent in a projective space, and with this, the

order

of a projective space may be defined: it is one less than the

cardinality

of a line in the space.

Without Axiom 3, we would end up with projective spaces such that some lines containing fewer than three points. These are either the degenerate cases where

I

is such that all three axioms are satisfied vacuously (for example, a point and a line with no incidence relation at all, etc..), or

structures

that are

disjoint union

of spaces (satisfying all three axioms) such that pairs of points from different spaces lie on a line consisting of exactly the pair of points.

Subspaces and Dimensions

What seems to be absent in the axioms of projective geometry is any mentioning of another fundamental notion: plane. In fact, a plane can be defined using points, lines, and the incidence relation: a plane is the smallest set

P

of points (in the projective space) such that there is a line

ℓ

⊆

P

and a point

p

∈

P

not on

ℓ

, such that

p

⁢

q

⊆

P

for any

q

on

ℓ

. We may write

P

=

⟨

p

,

ℓ

⟩

, and say that

P

is determined by

p

and

ℓ

. It is not hard to see that any non-incident pair of a line and a point on

P

determine

P

. A plane in a projective space is also a model of a

projective plane

(but not all models of a projective plane are

isomorphic

!).

More generally, we may define a subspace of a projective space as a set

X

of points such that if

p

,

q

∈

X

, then

p

⁢

q

⊆

X

. Equivalently, a subspace of a projective space, viewed as an incidence structure, is an incidence

substructure

(with inherited incidence relation) that is also a projective space. Given a projective space, the

empty set

is a subspace, and so is any point (properly a one-point set), as well as any line, and any plane.

Dimensions

can also be defined on subspaces of a projective space, recursively:

∅

has dimension

-

1

, a point has dimension

0

, and if subspace

V

has dimension

n

, then any subspace containing

V

and any point not in

V

has dimension

n

+

1

. This process can go on, until the whole space itself is reached, in which case, we say the projective space is finite-dimensional. Or, it can go on indefinitely, in which case we say that the space is infinite-dimensional. But for an infinite dimensional projective space, what is its dimension, since there are different magnitudes of

infinity

. Veblen has shown the following:

any projective spaces of dimension

n

>

2

(including infinite dimensional ones) is isomorphic to

P

⁢

G

⁢

(

V

)

for some vector space

V

.

Thus, we may instead define the dimension of a projective space to be

dim

⁡

(

V

)

-

1

, where the space is isomorphic to

P

⁢

G

⁢

(

V

)

. This definition extends the

recursive

one given earlier. This also shows that if a projective space is infinite dimensional, its dimension is the same as the dimension of the vector space that generates it. A more direct way of defining dimension is to use the concept of

independent

spanning set

of a projective space,

similar

to that found for vector spaces.

References

1

A. Beutelspacher, U. Rosenbaum

Projective Geometry, From

Foundations

to Applications

, Cambridge University Press (2000)

2

O. Veblen, J. W. Young

Projective Geometry, Volumes I & II

, Ginn & Co., Boston, MA (1916)

Title

axiomatic projective geometry

Canonical name

AxiomaticProjectiveGeometry

Date of creation

2013-03-22 19:14:19

Last modified on

2013-03-22 19:14:19

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Definition

Classification

msc 05B35

Classification

msc 06C10

Classification

msc 51A05

Related topic

ProjectiveGeometry

Related topic

ProjectiveSpace

Related topic

ProjectivePlane2

Defines

Veblen’s axiom

Defines

order

Generated on Thu Feb 8 20:39:28 2018 by

LaTeXML