projective space

projective space

Projective space and homogeneous coordinates.

Let

𝕂

be a field.

Projective space

of

dimension

n

over

𝕂

, typically denoted by

𝕂

⁢

P

n

, is the set of lines

passing
through

the origin in

𝕂

n

+

1

. More formally, consider the

equivalence relation

∼

on the set of non-zero points

𝕂

n

+

1

\

{

0

}

defined by

𝐱

∼

λ

⁢

𝐱

,

𝐱

∈

𝕂

n

+

1

\

{

0

}

,

λ

∈

𝕂

\

{

0

}

.

Projective space is defined to be the set of the
corresponding

equivalence classes

.

Every

𝐱

=

(

x

0

,

…

,

x

n

)

∈

𝕂

n

+

1

\

{

0

}

determines an

element of

projective space, namely the line passing through

𝐱

. Formally,
this line is the equivalence class

[

𝐱

]

, or

[

x

0

:

x

1

:

…

:

x

n

]

,
as it is commonly denoted. The numbers

x

0

,

…

,

x

n

are referred
to as

homogeneous coordinates

of the line. Homogeneous coordinates
differ from ordinary

coordinate systems

in that a given element of
projective space is labeled by

multiple

homogeneous

“

coordinates

”.

Affine coordinates.

Projective space also admits a more
conventional type of coordinate system, called

affine coordinates

.
Let

A

0

⊂

𝕂

⁢

P

n

be the subset of all

elements

p

=

[

x

0

:

x

1

:

…

:

x

n

]

∈

𝕂

P

n

such that

x

0

≠

0

. We then
define the

functions

X

i

:

A

0

→

𝕂

n

,

i

=

1

,

…

,

n

,

according to

X

i

⁢

(

p

)

=

x

i

x

0

,

where

(

x

0

,

x

1

,

…

,

x

n

)

is

any

element of the equivalence class representing

p

. This
definition makes sense because other elements of the same equivalence
class have the form

(

y

0

,

y

1

,

…

,

y

n

)

=

(

λ

⁢

x

0

,

λ

⁢

x

1

,

…

,

λ

⁢

x

n

)

for some non-zero

λ

∈

𝕂

, and hence

y

i

y

0

=

x

i

x

0

.

The functions

X

1

,

…

,

X

n

are called

affine coordinates

relative
to the

hyperplane

H

0

=

{

x

0

=

1

}

⊂

𝕂

n

+

1

.

Geometrically,
affine coordinates can be described by saying that the elements of

A

0

are lines in

𝕂

n

+

1

that are not

parallel

to

H

0

, and
that every such line

intersects

H

0

in one and exactly one point.

Conversely

points of

H

0

are represented by tuples

(

1

,

x

1

,

…

,

x

n

)

with

(

x

1

,

…

,

x

n

)

∈

𝕂

n

, and each such
point uniquely labels a line

[

1

:

x

1

:

…

:

x

n

]

in

A

0

.

It must be noted that a single system of affine coordinates does not
cover

all

of projective space. However, it is possible to
define a system of affine coordinates relative to every hyperplane in

𝕂

n

+

1

that does not contain the origin. In particular, we get

n

+

1

different systems of affine coordinates corresponding to the
hyperplanes

{

x

i

=

1

}

,

i

=

0

,

1

,

…

,

n

.

Every element of
projective space is covered by at least one of these

n

+

1

systems of
coordinates.

Projective automorphisms.

A projective automorphism, also known as a

projectivity

, is a

bijective

transformation

of projective space that preserves all

incidence relations

. For

n

≥

2

, every

automorphism

of

𝕂

⁢

P

n

is
engendered by a semilinear

invertible

transformation of

𝕂

n

+

1

.
Let

A

:

𝕂

n

+

1

→

𝕂

n

+

1

be an invertible

semilinear
transformation

. The corresponding projectivity

[

A

]

:

𝕂

⁢

P

n

→

𝕂

⁢

P

n

is the transformation

[

𝐱

]

↦

[

A

⁢

𝐱

]

,

𝐱

∈

𝕂

n

+

1

.

For every non-zero

λ

∈

𝕂

the
transformation

λ

⁢

A

gives the same projective automorphism as

A

. For this reason, it is convenient we identify the group of
projective automorphisms with the

quotient

P

⁢

Γ

⁢

L

n

+

1

⁡

(

𝕂

)

=

Γ

⁢

L

n

+

1

⁡

(

𝕂

)

/

𝕂

.

Here

Γ

⁢

L

refers to the group of
invertible semi-linear transformations, while the quotienting

𝕂

refers to the

subgroup

of

scalar multiplications

.

A

collineation

is a special kind of projective automorphism, one that
is engendered by a strictly

linear transformation

. The group of

projective collineations

is therefore denoted by

PGL

n

+

1

⁢

(

𝕂

)

Note that for fields such as

ℝ

and

ℂ

, the group of
projective collineations is also described by the projectivizations

PSL

n

+

1

⁢

(

ℝ

)

,

PSL

n

+

1

⁢

(

ℂ

)

, of the corresponding

unimodular group

.

Also note that if a field, such as

ℝ

, lacks non-trivial
automorphisms, then all semi-linear transformations are linear. For
such fields all projective automorphisms are collineations.
Thus,

P

⁢

Γ

⁢

L

n

+

1

⁡

(

ℝ

)

=

PSL

n

+

1

⁢

(

ℝ

)

=

SL

n

+

1

⁢

(

ℝ

)

/

{

±

I

n

+

1

}

.

By contrast, since

ℂ

possesses non-trivial automorphisms, complex conjugation for example,
the group of automorphisms of

complex projective space

is larger than

PSL

n

+

1

⁢

(

ℂ

)

, where the latter denotes the quotient of

SL

n

+

1

⁢

(

ℂ

)

by the subgroup of

scalings

by the

(

n

+

1

)

st roots
of unity.

Title

projective space

Canonical name

ProjectiveSpace

Date of creation

2013-03-22 12:03:53

Last modified on

2013-03-22 12:03:53

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

13

Author

rmilson (146)

Entry type

Definition

Classification

msc 14-00

Related topic

Projectivity

Related topic

SemilinearTransformation

Defines

homogeneous coordinates

Defines

affine coordinates

Defines

projective automorphism

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