<s>
In	O
projective	O
geometry	O
,	O
a	O
homography	B-Algorithm
is	O
an	O
isomorphism	O
of	O
projective	O
spaces	O
,	O
induced	O
by	O
an	O
isomorphism	O
of	O
the	O
vector	O
spaces	O
from	O
which	O
the	O
projective	O
spaces	O
derive	O
.	O
</s>
<s>
It	O
is	O
a	O
bijection	B-Algorithm
that	O
maps	O
lines	O
to	O
lines	O
,	O
and	O
thus	O
a	O
collineation	O
.	O
</s>
<s>
In	O
general	O
,	O
some	O
collineations	O
are	O
not	O
homographies	B-Algorithm
,	O
but	O
the	O
fundamental	O
theorem	O
of	O
projective	O
geometry	O
asserts	O
that	O
is	O
not	O
so	O
in	O
the	O
case	O
of	O
real	O
projective	O
spaces	O
of	O
dimension	O
at	O
least	O
two	O
.	O
</s>
<s>
Synonyms	O
include	O
projectivity	B-Algorithm
,	O
projective	B-Algorithm
transformation	I-Algorithm
,	O
and	O
projective	O
collineation	O
.	O
</s>
<s>
Historically	O
,	O
homographies	B-Algorithm
(	O
and	O
projective	O
spaces	O
)	O
have	O
been	O
introduced	O
to	O
study	O
perspective	B-Application
and	O
projections	O
in	O
Euclidean	O
geometry	O
,	O
and	O
the	O
term	O
homography	B-Algorithm
,	O
which	O
,	O
etymologically	O
,	O
roughly	O
means	O
"	O
similar	O
drawing	O
"	O
,	O
dates	O
from	O
this	O
time	O
.	O
</s>
<s>
The	O
term	O
"	O
projective	B-Algorithm
transformation	I-Algorithm
"	O
originated	O
in	O
these	O
abstract	O
constructions	O
.	O
</s>
<s>
A	O
projective	O
space	O
may	O
be	O
constructed	O
as	O
the	O
set	O
of	O
the	O
lines	O
of	O
a	O
vector	O
space	O
over	O
a	O
given	O
field	O
(	O
the	O
above	O
definition	O
is	O
based	O
on	O
this	O
version	O
)	O
;	O
this	O
construction	O
facilitates	O
the	O
definition	O
of	O
projective	O
coordinates	O
and	O
allows	O
using	O
the	O
tools	O
of	O
linear	B-Language
algebra	I-Language
for	O
the	O
study	O
of	O
homographies	B-Algorithm
.	O
</s>
<s>
The	O
alternative	O
approach	O
consists	O
in	O
defining	O
the	O
projective	O
space	O
through	O
a	O
set	O
of	O
axioms	O
,	O
which	O
do	O
not	O
involve	O
explicitly	O
any	O
field	O
(	O
incidence	O
geometry	O
,	O
see	O
also	O
synthetic	O
geometry	O
)	O
;	O
in	O
this	O
context	O
,	O
collineations	O
are	O
easier	O
to	O
define	O
than	O
homographies	B-Algorithm
,	O
and	O
homographies	B-Algorithm
are	O
defined	O
as	O
specific	O
collineations	O
,	O
thus	O
called	O
"	O
projective	O
collineations	O
"	O
.	O
</s>
<s>
Historically	O
,	O
the	O
concept	O
of	O
homography	B-Algorithm
had	O
been	O
introduced	O
to	O
understand	O
,	O
explain	O
and	O
study	O
visual	B-Architecture
perspective	I-Architecture
,	O
and	O
,	O
specifically	O
,	O
the	O
difference	O
in	O
appearance	O
of	O
two	O
plane	O
objects	O
viewed	O
from	O
different	O
points	O
of	O
view	O
.	O
</s>
<s>
Given	O
another	O
plane	O
Q	O
,	O
which	O
does	O
not	O
contain	O
O	O
,	O
the	O
restriction	O
to	O
Q	O
of	O
the	O
above	O
projection	O
is	O
called	O
a	O
perspectivity	B-Application
.	O
</s>
<s>
With	O
these	O
definitions	O
,	O
a	O
perspectivity	B-Application
is	O
only	O
a	O
partial	B-Algorithm
function	I-Algorithm
,	O
but	O
it	O
becomes	O
a	O
bijection	B-Algorithm
if	O
extended	O
to	O
projective	O
spaces	O
.	O
</s>
<s>
If	O
f	O
is	O
a	O
perspectivity	B-Application
from	O
P	O
to	O
Q	O
,	O
and	O
g	O
a	O
perspectivity	B-Application
from	O
Q	O
to	O
P	O
,	O
with	O
a	O
different	O
center	O
,	O
then	O
is	O
a	O
homography	B-Algorithm
from	O
P	O
to	O
itself	O
,	O
which	O
is	O
called	O
a	O
central	O
collineation	O
,	O
when	O
the	O
dimension	O
of	O
P	O
is	O
at	O
least	O
two	O
.	O
</s>
<s>
Originally	O
,	O
a	O
homography	B-Algorithm
was	O
defined	O
as	O
the	O
composition	B-Application
of	O
a	O
finite	O
number	O
of	O
perspectivities	B-Application
.	O
</s>
<s>
Given	O
two	O
projective	O
spaces	O
P(V )	O
and	O
P(W )	O
of	O
the	O
same	O
dimension	O
,	O
a	O
homography	B-Algorithm
is	O
a	O
mapping	O
from	O
P(V )	O
to	O
P(W )	O
,	O
which	O
is	O
induced	O
by	O
an	O
isomorphism	O
of	O
vector	O
spaces	O
.	O
</s>
<s>
Such	O
an	O
isomorphism	O
induces	O
a	O
bijection	B-Algorithm
from	O
P(V )	O
to	O
P(W )	O
,	O
because	O
of	O
the	O
linearity	O
of	O
f	O
.	O
Two	O
such	O
isomorphisms	O
,	O
f	O
and	O
g	O
,	O
define	O
the	O
same	O
homography	B-Algorithm
if	O
and	O
only	O
if	O
there	O
is	O
a	O
nonzero	O
element	O
a	O
of	O
K	O
such	O
that	O
.	O
</s>
<s>
This	O
may	O
be	O
written	O
in	O
terms	O
of	O
homogeneous	O
coordinates	O
in	O
the	O
following	O
way	O
:	O
A	O
homography	B-Algorithm
φ	O
may	O
be	O
defined	O
by	O
a	O
nonsingular	O
matrix	O
[ai,j],	O
called	O
the	O
matrix	O
of	O
the	O
homography	B-Algorithm
.	O
</s>
<s>
This	O
defines	O
only	O
a	O
partial	B-Algorithm
function	I-Algorithm
between	O
affine	O
spaces	O
,	O
which	O
is	O
defined	O
only	O
outside	O
the	O
hyperplane	O
where	O
the	O
denominator	O
is	O
zero	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
the	O
complex	O
projective	O
line	O
,	O
which	O
can	O
be	O
identified	O
with	O
the	O
Riemann	O
sphere	O
,	O
the	O
homographies	B-Algorithm
are	O
called	O
Möbius	O
transformations	O
.	O
</s>
<s>
These	O
correspond	O
precisely	O
with	O
those	O
bijections	B-Algorithm
of	O
the	O
Riemann	O
sphere	O
that	O
preserve	O
orientation	O
and	O
are	O
conformal	O
.	O
</s>
<s>
When	O
the	O
line	O
is	O
viewed	O
as	O
a	O
projective	O
space	O
in	O
isolation	O
,	O
any	O
permutation	B-Algorithm
of	O
the	O
points	O
of	O
a	O
projective	O
line	O
is	O
a	O
collineation	O
,	O
since	O
every	O
set	O
of	O
points	O
are	O
collinear	O
.	O
</s>
<s>
Thus	O
,	O
in	O
synthetic	O
geometry	O
,	O
the	O
homographies	B-Algorithm
and	O
the	O
collineations	O
of	O
the	O
projective	O
line	O
that	O
are	O
considered	O
are	O
those	O
obtained	O
by	O
restrictions	O
to	O
the	O
line	O
of	O
collineations	O
and	O
homographies	B-Algorithm
of	O
spaces	O
of	O
higher	O
dimension	O
.	O
</s>
<s>
A	O
homography	B-Algorithm
of	O
a	O
projective	O
line	O
may	O
also	O
be	O
properly	O
defined	O
by	O
insisting	O
that	O
the	O
mapping	O
preserves	O
cross-ratios	O
.	O
</s>
<s>
It	O
follows	O
that	O
,	O
given	O
two	O
frames	O
,	O
there	O
is	O
exactly	O
one	O
homography	B-Algorithm
mapping	O
the	O
first	O
one	O
onto	O
the	O
second	O
one	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
only	O
homography	B-Algorithm
fixing	O
the	O
points	O
of	O
a	O
frame	O
is	O
the	O
identity	O
map	O
.	O
</s>
<s>
On	O
this	O
basis	O
,	O
the	O
homogeneous	O
coordinates	O
of	O
are	O
simply	O
the	O
entries	O
(	O
coefficients	O
)	O
of	O
the	O
tuple	B-Application
.	O
</s>
<s>
Given	O
another	O
projective	O
space	O
of	O
the	O
same	O
dimension	O
,	O
and	O
a	O
frame	O
of	O
it	O
,	O
there	O
is	O
one	O
and	O
only	O
one	O
homography	B-Algorithm
mapping	O
onto	O
the	O
canonical	O
frame	O
of	O
.	O
</s>
<s>
In	O
above	O
sections	O
,	O
homographies	B-Algorithm
have	O
been	O
defined	O
through	O
linear	B-Language
algebra	I-Language
.	O
</s>
<s>
In	O
synthetic	O
geometry	O
,	O
they	O
are	O
traditionally	O
defined	O
as	O
the	O
composition	B-Application
of	O
one	O
or	O
several	O
special	O
homographies	B-Algorithm
called	O
central	O
collineations	O
.	O
</s>
<s>
In	O
a	O
projective	O
space	O
,	O
P	O
,	O
of	O
dimension	O
,	O
a	O
collineation	O
of	O
P	O
is	O
a	O
bijection	B-Algorithm
from	O
P	O
onto	O
P	O
that	O
maps	O
lines	O
onto	O
lines	O
.	O
</s>
<s>
A	O
central	O
collineation	O
(	O
traditionally	O
these	O
were	O
called	O
perspectivities	B-Application
,	O
but	O
this	O
term	O
may	O
be	O
confusing	O
,	O
having	O
another	O
meaning	O
;	O
see	O
Perspectivity	B-Application
)	O
is	O
a	O
bijection	B-Algorithm
α	O
from	O
P	O
to	O
P	O
,	O
such	O
that	O
there	O
exists	O
a	O
hyperplane	O
H	O
(	O
called	O
the	O
axis	O
of	O
α	O
)	O
,	O
which	O
is	O
fixed	O
pointwise	O
by	O
α	O
(	O
that	O
is	O
,	O
for	O
all	O
points	O
X	O
in	O
H	O
)	O
and	O
a	O
point	O
O	O
(	O
called	O
the	O
center	O
of	O
α	O
)	O
,	O
which	O
is	O
fixed	O
linewise	O
by	O
α	O
(	O
any	O
line	O
through	O
O	O
is	O
mapped	O
to	O
itself	O
by	O
α	O
,	O
but	O
not	O
necessarily	O
pointwise	O
)	O
.	O
</s>
<s>
A	O
central	O
collineation	O
is	O
a	O
homography	B-Algorithm
defined	O
by	O
a	O
(	O
n+1	O
)	O
×	O
(	O
n+1	O
)	O
matrix	O
that	O
has	O
an	O
eigenspace	O
of	O
dimension	O
n	O
.	O
It	O
is	O
a	O
homology	O
,	O
if	O
the	O
matrix	O
has	O
another	O
eigenvalue	O
and	O
is	O
therefore	O
diagonalizable	B-Algorithm
.	O
</s>
<s>
It	O
is	O
an	O
elation	O
,	O
if	O
all	O
the	O
eigenvalues	O
are	O
equal	O
and	O
the	O
matrix	O
is	O
not	O
diagonalizable	B-Algorithm
.	O
</s>
<s>
The	O
composition	B-Application
of	O
two	O
central	O
collineations	O
,	O
while	O
still	O
a	O
homography	B-Algorithm
in	O
general	O
,	O
is	O
not	O
a	O
central	O
collineation	O
.	O
</s>
<s>
In	O
fact	O
,	O
every	O
homography	B-Algorithm
is	O
the	O
composition	B-Application
of	O
a	O
finite	O
number	O
of	O
central	O
collineations	O
.	O
</s>
<s>
In	O
synthetic	O
geometry	O
,	O
this	O
property	O
,	O
which	O
is	O
a	O
part	O
of	O
the	O
fundamental	O
theory	O
of	O
projective	O
geometry	O
is	O
taken	O
as	O
the	O
definition	O
of	O
homographies	B-Algorithm
.	O
</s>
<s>
There	O
are	O
collineations	O
besides	O
the	O
homographies	B-Algorithm
.	O
</s>
<s>
Given	O
two	O
projective	O
frames	O
of	O
a	O
projective	O
space	O
P	O
,	O
there	O
is	O
exactly	O
one	O
homography	B-Algorithm
of	O
P	O
that	O
maps	O
the	O
first	O
frame	O
onto	O
the	O
second	O
one	O
.	O
</s>
<s>
If	O
the	O
dimension	O
of	O
a	O
projective	O
space	O
P	O
is	O
at	O
least	O
two	O
,	O
every	O
collineation	O
of	O
P	O
is	O
the	O
composition	B-Application
of	O
an	O
automorphic	O
collineation	O
and	O
a	O
homography	B-Algorithm
.	O
</s>
<s>
In	O
particular	O
,	O
over	O
the	O
reals	O
,	O
every	O
collineation	O
of	O
a	O
projective	O
space	O
of	O
dimension	O
at	O
least	O
two	O
is	O
a	O
homography	B-Algorithm
.	O
</s>
<s>
Every	O
homography	B-Algorithm
is	O
the	O
composition	B-Application
of	O
a	O
finite	O
number	O
of	O
perspectivities	B-Application
.	O
</s>
<s>
In	O
particular	O
,	O
if	O
the	O
dimension	O
of	O
the	O
implied	O
projective	O
space	O
is	O
at	O
least	O
two	O
,	O
every	O
homography	B-Algorithm
is	O
the	O
composition	B-Application
of	O
a	O
finite	O
number	O
of	O
central	O
collineations	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
if	O
projective	O
spaces	O
are	O
defined	O
by	O
means	O
of	O
linear	B-Language
algebra	I-Language
,	O
the	O
first	O
part	O
is	O
an	O
easy	O
corollary	O
of	O
the	O
definitions	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
proof	O
of	O
the	O
first	O
part	O
in	O
synthetic	O
geometry	O
,	O
and	O
the	O
proof	O
of	O
the	O
third	O
part	O
in	O
terms	O
of	O
linear	B-Language
algebra	I-Language
both	O
are	O
fundamental	O
steps	O
of	O
the	O
proof	O
of	O
the	O
equivalence	O
of	O
the	O
two	O
ways	O
of	O
defining	O
projective	O
spaces	O
.	O
</s>
<s>
As	O
every	O
homography	B-Algorithm
has	O
an	O
inverse	O
mapping	O
and	O
the	O
composition	B-Application
of	O
two	O
homographies	B-Algorithm
is	O
another	O
,	O
the	O
homographies	B-Algorithm
of	O
a	O
given	O
projective	O
space	O
form	O
a	O
group	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
Möbius	O
group	O
is	O
the	O
homography	B-Algorithm
group	O
of	O
any	O
complex	O
projective	O
line	O
.	O
</s>
<s>
As	O
all	O
the	O
projective	O
spaces	O
of	O
the	O
same	O
dimension	O
over	O
the	O
same	O
field	O
are	O
isomorphic	O
,	O
the	O
same	O
is	O
true	O
for	O
their	O
homography	B-Algorithm
groups	O
.	O
</s>
<s>
Homography	B-Algorithm
groups	O
also	O
called	O
projective	O
linear	O
groups	O
are	O
denoted	O
when	O
acting	O
on	O
a	O
projective	O
space	O
of	O
dimension	O
n	O
over	O
a	O
field	O
F	O
.	O
Above	O
definition	O
of	O
homographies	B-Algorithm
shows	O
that	O
may	O
be	O
identified	O
to	O
the	O
quotient	O
group	O
,	O
where	O
is	O
the	O
general	O
linear	O
group	O
of	O
the	O
invertible	O
matrices	O
,	O
and	O
F×I	O
is	O
the	O
group	O
of	O
the	O
products	O
by	O
a	O
nonzero	O
element	O
of	O
F	O
of	O
the	O
identity	O
matrix	O
of	O
size	O
.	O
</s>
<s>
When	O
F	O
is	O
a	O
Galois	O
field	O
GF(q )	O
then	O
the	O
homography	B-Algorithm
group	O
is	O
written	O
.	O
</s>
<s>
For	O
example	O
,	O
acts	O
on	O
the	O
eight	O
points	O
in	O
the	O
projective	O
line	O
over	O
the	O
finite	O
field	O
GF(7 )	O
,	O
while	O
,	O
which	O
is	O
isomorphic	O
to	O
the	O
alternating	B-Algorithm
group	I-Algorithm
A5	O
,	O
is	O
the	O
homography	B-Algorithm
group	O
of	O
the	O
projective	O
line	O
with	O
five	O
points	O
.	O
</s>
<s>
The	O
homography	B-Algorithm
group	O
is	O
a	O
subgroup	O
of	O
the	O
collineation	O
group	O
of	O
the	O
collineations	O
of	O
a	O
projective	O
space	O
of	O
dimension	O
n	O
.	O
When	O
the	O
points	O
and	O
lines	O
of	O
the	O
projective	O
space	O
are	O
viewed	O
as	O
a	O
block	O
design	O
,	O
whose	O
blocks	O
are	O
the	O
sets	O
of	O
points	O
contained	O
in	O
a	O
line	O
,	O
it	O
is	O
common	O
to	O
call	O
the	O
collineation	O
group	O
the	O
automorphism	O
group	O
of	O
the	O
design	O
.	O
</s>
<s>
The	O
cross-ratio	O
of	O
four	O
collinear	O
points	O
is	O
an	O
invariant	O
under	O
the	O
homography	B-Algorithm
that	O
is	O
fundamental	O
for	O
the	O
study	O
of	O
the	O
homographies	B-Algorithm
of	O
the	O
lines	O
.	O
</s>
<s>
There	O
is	O
therefore	O
a	O
unique	O
homography	B-Algorithm
of	O
this	O
line	O
onto	O
that	O
maps	O
to	O
,	O
to	O
0	O
,	O
and	O
to	O
1	O
.	O
</s>
<s>
Homographies	B-Algorithm
act	O
on	O
a	O
projective	O
line	O
over	O
A	O
,	O
written	O
P(A )	O
,	O
consisting	O
of	O
points	O
with	O
projective	O
coordinates	O
.	O
</s>
<s>
The	O
homography	B-Algorithm
group	O
of	O
the	O
ring	O
of	O
integers	O
Z	O
is	O
modular	O
group	O
.	O
</s>
<s>
Ring	O
homographies	B-Algorithm
have	O
been	O
used	O
in	O
quaternion	B-Algorithm
analysis	I-Algorithm
,	O
and	O
with	O
dual	O
quaternions	O
to	O
facilitate	O
screw	O
theory	O
.	O
</s>
<s>
The	O
conformal	O
group	O
of	O
spacetime	O
can	O
be	O
represented	O
with	O
homographies	B-Algorithm
where	O
A	O
is	O
the	O
composition	B-Application
algebra	O
of	O
biquaternions	O
.	O
</s>
<s>
In	O
his	O
review	O
of	O
a	O
brute	O
force	O
approach	O
to	O
periodicity	O
of	O
homographies	B-Algorithm
,	O
H	O
.	O
S	O
.	O
M	O
.	O
Coxeter	O
gave	O
this	O
analysis	O
:	O
</s>
<s>
A	O
real	O
homography	B-Algorithm
is	O
involutory	O
(	O
of	O
period	O
2	O
)	O
if	O
and	O
only	O
if	O
.	O
</s>
