projectivity

projectivity

Let

P

⁢

G

⁢

(

V

)

and

P

⁢

G

⁢

(

W

)

be

projective geometries

, with

V

,

W

vector spaces

over a field

K

. A

function

p

from

P

⁢

G

⁢

(

V

)

to

P

⁢

G

⁢

(

W

)

is called a

projective transformation

, or simply a

projectivity

if

1.

p

is a

bijection

, and

2.

p

is order preserving.

A

projective property

is any geometric property, such as incidence, linearity, etc… that is preserved under a projectivity.

From the definition, we see that a projectivity

p

carries 0 to 0,

V

to

W

. Furthermore, it carries points to points, lines to lines, planes to planes, etc.. In short,

p

preserves linearity. Because

p

is a bijection,

p

also preserves

dimensions

, that is

dim

⁡

(

S

)

=

dim

⁡

(

p

⁢

(

S

)

)

, for any

subspace

S

of

V

. In particular,

dim

⁡

(

V

)

=

dim

⁡

(

W

)

. Other properties preserved by

p

are incidence: if

S

∩

T

≠

∅

, then

p

⁢

(

S

)

∩

p

⁢

(

T

)

≠

∅

; and cross ratios (

http://planetmath.org/CrossRatio

).

Every

bijective

semilinear transformation

defines a projectiviity. To see this, let

f

:

V

→

W

be a semilinear transformation. If

S

is a subspace of

V

, then

f

⁢

(

S

)

is a subspace of

W

, as

x

,

y

∈

f

⁢

(

S

)

, then

x

+

y

=

f

⁢

(

a

)

+

f

⁢

(

b

)

=

f

⁢

(

a

+

b

)

∈

f

⁢

(

S

)

, and

α

⁢

x

=

β

θ

⁢

x

=

β

θ

⁢

f

⁢

(

a

)

=

f

⁢

(

β

⁢

a

)

∈

f

⁢

(

S

)

, where

θ

is an

automorphism

of the common underlying field

K

. Also, if

S

is a subspace of a subspace

T

of

V

, then

f

⁢

(

S

)

is a subspace of

f

⁢

(

T

)

. Now if we define

f

*

:

P

⁢

G

⁢

(

V

)

→

P

⁢

G

⁢

(

W

)

by

f

*

⁢

(

S

)

=

f

⁢

(

S

)

, it is easy to see that

f

*

is a projectivity.

Conversely

, if

V

and

W

are of finite dimension greater than

2

, then a projectivity

p

:

P

⁢

G

⁢

(

V

)

→

P

⁢

G

⁢

(

W

)

induces a semilinear transformation

p

^

:

V

→

W

. This highly non-trivial fact is the (first)

fundamental theorem of projective geometry

.

If the semilinear transformation induced by the projectivity

p

is in fact a

linear transformation

, then

p

is a

collineation

: three distinct

collinear points

are mapped to three distinct collinear points.

Remark

. The definition given in this entry is a

generalization

of the definition typically given for a projective transformation. In the more restictive definition, a projectivity

p

is defined merely as a bijection between two

projective spaces

that is induced by a

linear isomorphism

. More precisely, if

P

⁢

(

V

)

and

P

⁢

(

W

)

are projective spaces induced by the vector spaces

V

and

W

, if

L

:

V

→

W

is a bijective linear transformation, then

p

=

P

⁢

(

L

)

:

P

⁢

(

V

)

→

P

⁢

(

W

)

defined by

P

⁢

(

L

)

⁢

[

v

]

=

[

L

⁢

v

]

is the corresponding projective transformation.

[

v

]

is the

homogeneous coordinate

representation

of

v

. In this definition, a projectiity is always a collineation. In the case where the vector spaces are finite dimensional with specified bases,

p

is expressible in terms of an

invertible matrix

(

L

⁢

v

=

A

⁢

v

where

A

is an invertible matrix).

Title

projectivity

Canonical name

Projectivity

Date of creation

2013-03-22 15:58:00

Last modified on

2013-03-22 15:58:00

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51A10

Classification

msc 51A05

Related topic

PolaritiesAndForms

Related topic

SesquilinearFormsOverGeneralFields

Related topic

Perspectivity

Related topic

ProjectiveSpace

Related topic

LinearFunction

Related topic

Collineation

Defines

projective transformation

Defines

projective property

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