<s>
In	O
geometry	O
and	O
in	O
its	O
applications	O
to	O
drawing	O
,	O
a	O
perspectivity	B-Application
is	O
the	O
formation	O
of	O
an	O
image	O
in	O
a	O
picture	O
plane	O
of	O
a	O
scene	O
viewed	O
from	O
a	O
fixed	O
point	O
.	O
</s>
<s>
The	O
science	O
of	O
graphical	B-Architecture
perspective	I-Architecture
uses	O
perspectivities	B-Application
to	O
make	O
realistic	O
images	O
in	O
proper	O
proportion	O
.	O
</s>
<s>
According	O
to	O
Kirsti	O
Andersen	O
,	O
the	O
first	O
author	O
to	O
describe	O
perspectivity	B-Application
was	O
Leon	O
Alberti	O
in	O
his	O
De	O
Pictura	O
(	O
1435	O
)	O
.	O
</s>
<s>
In	O
English	O
,	O
Brook	O
Taylor	O
presented	O
his	O
Linear	B-Architecture
Perspective	I-Architecture
in	O
1715	O
,	O
where	O
he	O
explained	O
"	O
Perspective	O
is	O
the	O
Art	O
of	O
drawing	O
on	O
a	O
Plane	O
the	O
Appearances	O
of	O
any	O
Figures	O
,	O
by	O
the	O
Rules	O
of	O
Geometry	O
"	O
.	O
</s>
<s>
Given	O
two	O
lines	O
and	O
in	O
a	O
projective	O
plane	O
and	O
a	O
point	O
P	O
of	O
that	O
plane	O
on	O
neither	O
line	O
,	O
the	O
bijective	B-Algorithm
mapping	I-Algorithm
between	O
the	O
points	O
of	O
the	O
range	O
of	O
and	O
the	O
range	O
of	O
determined	O
by	O
the	O
lines	O
of	O
the	O
pencil	O
on	O
P	O
is	O
called	O
a	O
perspectivity	B-Application
(	O
or	O
more	O
precisely	O
,	O
a	O
central	O
perspectivity	B-Application
with	O
center	O
P	O
)	O
.	O
</s>
<s>
The	O
existence	O
of	O
a	O
perspectivity	B-Application
means	O
that	O
corresponding	O
points	O
are	O
in	O
perspective	O
.	O
</s>
<s>
The	O
dual	O
concept	O
,	O
axial	O
perspectivity	B-Application
,	O
is	O
the	O
correspondence	O
between	O
the	O
lines	O
of	O
two	O
pencils	O
determined	O
by	O
a	O
projective	O
range	O
.	O
</s>
<s>
The	O
composition	O
of	O
two	O
perspectivities	B-Application
is	O
,	O
in	O
general	O
,	O
not	O
a	O
perspectivity	B-Application
.	O
</s>
<s>
A	O
perspectivity	B-Application
or	O
a	O
composition	O
of	O
two	O
or	O
more	O
perspectivities	B-Application
is	O
called	O
a	O
projectivity	O
(	O
projective	O
transformation	O
,	O
projective	O
collineation	O
and	O
homography	B-Algorithm
are	O
synonyms	B-Application
)	O
.	O
</s>
<s>
There	O
are	O
several	O
results	O
concerning	O
projectivities	O
and	O
perspectivities	B-Application
which	O
hold	O
in	O
any	O
pappian	O
projective	O
plane	O
:	O
</s>
<s>
Theorem	O
:	O
Any	O
projectivity	O
between	O
two	O
distinct	O
projective	O
ranges	O
can	O
be	O
written	O
as	O
the	O
composition	O
of	O
no	O
more	O
than	O
two	O
perspectivities	B-Application
.	O
</s>
<s>
Theorem	O
:	O
Any	O
projectivity	O
from	O
a	O
projective	O
range	O
to	O
itself	O
can	O
be	O
written	O
as	O
the	O
composition	O
of	O
three	O
perspectivities	B-Application
.	O
</s>
<s>
Theorem	O
:	O
A	O
projectivity	O
between	O
two	O
distinct	O
projective	O
ranges	O
which	O
fixes	O
a	O
point	O
is	O
a	O
perspectivity	B-Application
.	O
</s>
<s>
The	O
bijective	B-Algorithm
correspondence	O
between	O
points	O
on	O
two	O
lines	O
in	O
a	O
plane	O
determined	O
by	O
a	O
point	O
of	O
that	O
plane	O
not	O
on	O
either	O
line	O
has	O
higher-dimensional	O
analogues	O
which	O
will	O
also	O
be	O
called	O
perspectivities	B-Application
.	O
</s>
<s>
This	O
correspondence	O
fP	O
is	O
also	O
called	O
a	O
perspectivity	B-Application
.	O
</s>
<s>
The	O
central	O
perspectivity	B-Application
described	O
above	O
is	O
the	O
case	O
with	O
and	O
.	O
</s>
<s>
With	O
O	O
and	O
O*	O
being	O
points	O
of	O
R3	O
in	O
neither	O
plane	O
,	O
use	O
the	O
construction	O
of	O
the	O
last	O
section	O
to	O
project	O
S2	O
onto	O
T2	O
by	O
the	O
perspectivity	B-Application
with	O
center	O
O	O
followed	O
by	O
the	O
projection	O
of	O
T2	O
back	O
onto	O
S2	O
with	O
the	O
perspectivity	B-Application
with	O
center	O
O*	O
.	O
</s>
<s>
This	O
composition	O
is	O
a	O
bijective	B-Algorithm
map	I-Algorithm
of	O
the	O
points	O
of	O
S2	O
onto	O
itself	O
which	O
preserves	O
collinear	O
points	O
and	O
is	O
called	O
a	O
perspective	O
collineation	O
(	O
central	O
collineation	O
in	O
more	O
modern	O
terminology	O
)	O
.	O
</s>
<s>
The	O
restriction	O
of	O
φ	O
to	O
any	O
line	O
of	O
S2	O
not	O
passing	O
through	O
P	O
is	O
the	O
central	O
perspectivity	B-Application
in	O
S2	O
with	O
center	O
P	O
between	O
that	O
line	O
and	O
the	O
line	O
which	O
is	O
its	O
image	O
under	O
φ	O
.	O
</s>
