<s>
which	O
is	O
a	O
uniform	B-Algorithm
scaling	I-Algorithm
and	O
shows	O
the	O
meaning	O
of	O
special	O
choices	O
for	O
:	O
</s>
<s>
In	O
Euclidean	O
geometry	O
homotheties	B-Algorithm
are	O
the	O
similarities	O
that	O
fix	O
a	O
point	O
and	O
either	O
preserve	O
(	O
if	O
)	O
or	O
reverse	O
(	O
if	O
)	O
the	O
direction	O
of	O
all	O
vectors	O
.	O
</s>
<s>
Together	O
with	O
the	O
translations	B-Algorithm
,	O
all	O
homotheties	B-Algorithm
of	O
an	O
affine	O
(	O
or	O
Euclidean	O
)	O
space	O
form	O
a	O
group	O
,	O
the	O
group	O
of	O
dilations	B-Algorithm
or	O
homothety-translations	O
.	O
</s>
<s>
These	O
are	O
precisely	O
the	O
affine	B-Algorithm
transformations	I-Algorithm
with	O
the	O
property	O
that	O
the	O
image	O
of	O
every	O
line	O
g	O
is	O
a	O
line	O
parallel	O
to	O
g	O
.	O
</s>
<s>
In	O
projective	O
geometry	O
,	O
a	O
homothetic	B-Algorithm
transformation	I-Algorithm
is	O
a	O
similarity	O
transformation	B-Algorithm
(	O
i.e.	O
,	O
fixes	O
a	O
given	O
elliptic	O
involution	O
)	O
that	O
leaves	O
the	O
line	O
at	O
infinity	O
pointwise	O
invariant	O
.	O
</s>
<s>
In	O
Euclidean	O
geometry	O
,	O
a	O
homothety	B-Algorithm
of	O
ratio	O
multiplies	O
distances	O
between	O
points	O
by	O
,	O
areas	O
by	O
and	O
volumes	O
by	O
.	O
</s>
<s>
Here	O
is	O
the	O
ratio	O
of	O
magnification	O
or	O
dilation	B-Algorithm
factor	O
or	O
scale	O
factor	O
or	O
similitude	O
ratio	O
.	O
</s>
<s>
Such	O
a	O
transformation	B-Algorithm
can	O
be	O
called	O
an	O
enlargement	B-Algorithm
if	O
the	O
scale	O
factor	O
exceeds1	O
.	O
</s>
<s>
Homotheties	B-Algorithm
are	O
used	O
to	O
scale	O
the	O
contents	O
of	O
computer	O
screens	O
;	O
for	O
example	O
,	O
smartphones	O
,	O
notebooks	O
,	O
and	O
laptops	O
.	O
</s>
<s>
A	O
homothety	B-Algorithm
has	O
the	O
following	O
properties	O
:	O
</s>
<s>
A	O
homothety	B-Algorithm
is	O
a	O
similarity	O
.	O
</s>
<s>
If	O
for	O
a	O
homothety	B-Algorithm
with	O
center	O
the	O
image	O
of	O
a	O
point	O
is	O
given	O
(	O
see	O
diagram	O
)	O
then	O
the	O
image	O
of	O
a	O
second	O
point	O
,	O
which	O
lies	O
not	O
on	O
line	O
can	O
be	O
constructed	O
graphically	O
using	O
the	O
intercept	O
theorem	O
:	O
is	O
the	O
common	O
point	O
th	O
two	O
lines	O
and	O
.	O
</s>
<s>
Before	O
computers	O
became	O
ubiquitous	O
,	O
scalings	B-Algorithm
of	O
drawings	O
were	O
done	O
by	O
using	O
a	O
pantograph	O
,	O
a	O
tool	O
similar	O
to	O
a	O
compass	O
.	O
</s>
<s>
That	O
shows	O
:	O
the	O
mapping	O
is	O
a	O
homothety	B-Algorithm
with	O
center	O
and	O
ratio	O
.	O
</s>
<s>
The	O
composition	O
of	O
two	O
homotheties	B-Algorithm
with	O
the	O
same	O
center	O
is	O
again	O
a	O
homothety	B-Algorithm
with	O
center	O
.	O
</s>
<s>
The	O
homotheties	B-Algorithm
with	O
center	O
form	O
a	O
group	O
.	O
</s>
<s>
in	O
case	O
of	O
a	O
translation	B-Algorithm
in	O
direction	O
.	O
</s>
<s>
Especially	O
,	O
if	O
(	O
point	B-Algorithm
reflections	I-Algorithm
)	O
.	O
</s>
<s>
in	O
case	O
of	O
a	O
translation	B-Algorithm
in	O
direction	O
by	O
vector	O
.	O
</s>
<s>
is	O
a	O
homothety	B-Algorithm
with	O
center	O
and	O
ratio	O
.	O
</s>
<s>
The	O
composition	O
of	O
a	O
homothety	B-Algorithm
and	O
a	O
translation	B-Algorithm
is	O
a	O
homothety	B-Algorithm
.	O
</s>
<s>
which	O
is	O
a	O
homothety	B-Algorithm
with	O
center	O
and	O
ratio	O
.	O
</s>
<s>
with	O
center	O
can	O
be	O
written	O
as	O
the	O
composition	O
of	O
a	O
homothety	B-Algorithm
with	O
center	O
and	O
a	O
translation	B-Algorithm
:	O
</s>
