<s>
In	O
mathematics	O
,	O
a	O
geometric	B-Algorithm
transformation	I-Algorithm
is	O
any	O
bijection	B-Algorithm
of	O
a	O
set	O
to	O
itself	O
(	O
or	O
to	O
another	O
such	O
set	O
)	O
with	O
some	O
salient	O
geometrical	O
underpinning	O
.	O
</s>
<s>
More	O
specifically	O
,	O
it	O
is	O
a	O
function	O
whose	O
domain	B-Algorithm
and	O
range	B-Algorithm
are	O
sets	O
of	O
points	O
—	O
most	O
often	O
both	O
or	O
both	O
—	O
such	O
that	O
the	O
function	O
is	O
bijective	B-Algorithm
so	O
that	O
its	O
inverse	O
exists	O
.	O
</s>
<s>
Geometric	B-Algorithm
transformations	I-Algorithm
can	O
be	O
classified	O
by	O
the	O
dimension	O
of	O
their	O
operand	O
sets	O
(	O
thus	O
distinguishing	O
between	O
,	O
say	O
,	O
planar	O
transformations	O
and	O
spatial	O
transformations	O
)	O
.	O
</s>
<s>
Displacements	O
preserve	O
distances	O
and	O
oriented	O
angles	O
(	O
e.g.	O
,	O
translations	B-Algorithm
)	O
;	O
</s>
<s>
Isometries	O
preserve	O
angles	O
and	O
distances	O
(	O
e.g.	O
,	O
Euclidean	B-Algorithm
transformations	I-Algorithm
)	O
;	O
</s>
<s>
Affine	B-Algorithm
transformations	I-Algorithm
preserve	O
parallelism	O
(	O
e.g.	O
,	O
scaling	B-Algorithm
,	O
shear	B-Algorithm
)	O
;	O
</s>
<s>
Projective	B-Algorithm
transformations	I-Algorithm
preserve	O
collinearity	O
;	O
</s>
<s>
Equiareal	B-Algorithm
transformations	I-Algorithm
,	O
preserve	O
areas	O
in	O
the	O
planar	O
case	O
or	O
volumes	O
in	O
the	O
three	O
dimensional	O
case	O
.	O
</s>
<s>
and	O
are	O
,	O
in	O
the	O
first	O
order	O
,	O
affine	B-Algorithm
transformations	I-Algorithm
of	O
determinant	O
1	O
.	O
</s>
<s>
Many	O
geometric	B-Algorithm
transformations	I-Algorithm
are	O
expressed	O
with	O
linear	O
algebra	O
.	O
</s>
<s>
The	O
bijective	B-Algorithm
linear	B-Architecture
transformations	I-Architecture
are	O
elements	O
of	O
a	O
general	O
linear	O
group	O
.	O
</s>
<s>
The	O
linear	B-Architecture
transformation	I-Architecture
A	O
is	O
non-singular	O
.	O
</s>
