<s>
In	O
affine	O
geometry	O
,	O
uniform	B-Algorithm
scaling	I-Algorithm
(	O
or	O
isotropic	B-Algorithm
scaling	I-Algorithm
)	O
is	O
a	O
linear	B-Architecture
transformation	I-Architecture
that	O
enlarges	O
(	O
increases	O
)	O
or	O
shrinks	O
(	O
diminishes	O
)	O
objects	O
by	O
a	O
scale	B-Algorithm
factor	I-Algorithm
that	O
is	O
the	O
same	O
in	O
all	O
directions	O
.	O
</s>
<s>
The	O
result	O
of	O
uniform	B-Algorithm
scaling	I-Algorithm
is	O
similar	O
(	O
in	O
the	O
geometric	O
sense	O
)	O
to	O
the	O
original	O
.	O
</s>
<s>
A	O
scale	B-Algorithm
factor	I-Algorithm
of	O
1	O
is	O
normally	O
allowed	O
,	O
so	O
that	O
congruent	O
shapes	O
are	O
also	O
classed	O
as	O
similar	O
.	O
</s>
<s>
Uniform	B-Algorithm
scaling	I-Algorithm
happens	O
,	O
for	O
example	O
,	O
when	O
enlarging	O
or	O
reducing	O
a	O
photograph	O
,	O
or	O
when	O
creating	O
a	O
scale	O
model	O
of	O
a	O
building	O
,	O
car	O
,	O
airplane	O
,	O
etc	O
.	O
</s>
<s>
More	O
general	O
is	O
scaling	B-Algorithm
with	O
a	O
separate	O
scale	B-Algorithm
factor	I-Algorithm
for	O
each	O
axis	O
direction	O
.	O
</s>
<s>
Non-uniform	B-Algorithm
scaling	I-Algorithm
(	O
anisotropic	O
scaling	B-Algorithm
)	O
is	O
obtained	O
when	O
at	O
least	O
one	O
of	O
the	O
scaling	B-Algorithm
factors	O
is	O
different	O
from	O
the	O
others	O
;	O
a	O
special	O
case	O
is	O
directional	O
scaling	B-Algorithm
or	O
stretching	B-Algorithm
(	O
in	O
one	O
direction	O
)	O
.	O
</s>
<s>
Non-uniform	B-Algorithm
scaling	I-Algorithm
changes	O
the	O
shape	O
of	O
the	O
object	O
;	O
e.g.	O
</s>
<s>
a	O
square	O
may	O
change	O
into	O
a	O
rectangle	O
,	O
or	O
into	O
a	O
parallelogram	O
if	O
the	O
sides	O
of	O
the	O
square	O
are	O
not	O
parallel	O
to	O
the	O
scaling	B-Algorithm
axes	O
(	O
the	O
angles	O
between	O
lines	O
parallel	O
to	O
the	O
axes	O
are	O
preserved	O
,	O
but	O
not	O
all	O
angles	O
)	O
.	O
</s>
<s>
When	O
the	O
scale	B-Algorithm
factor	I-Algorithm
is	O
larger	O
than	O
1	O
,	O
(	O
uniform	O
or	O
non-uniform	O
)	O
scaling	B-Algorithm
is	O
sometimes	O
also	O
called	O
dilation	B-Algorithm
or	O
enlargement	B-Algorithm
.	O
</s>
<s>
When	O
the	O
scale	B-Algorithm
factor	I-Algorithm
is	O
a	O
positive	O
number	O
smaller	O
than	O
1	O
,	O
scaling	B-Algorithm
is	O
sometimes	O
also	O
called	O
contraction	O
or	O
reduction	O
.	O
</s>
<s>
In	O
the	O
most	O
general	O
sense	O
,	O
a	O
scaling	B-Algorithm
includes	O
the	O
case	O
in	O
which	O
the	O
directions	O
of	O
scaling	B-Algorithm
are	O
not	O
perpendicular	O
.	O
</s>
<s>
It	O
also	O
includes	O
the	O
case	O
in	O
which	O
one	O
or	O
more	O
scale	B-Algorithm
factors	I-Algorithm
are	O
equal	O
to	O
zero	O
(	O
projection	B-Algorithm
)	O
,	O
and	O
the	O
case	O
of	O
one	O
or	O
more	O
negative	O
scale	B-Algorithm
factors	I-Algorithm
(	O
a	O
directional	O
scaling	B-Algorithm
by	O
-1	O
is	O
equivalent	O
to	O
a	O
reflection	B-Algorithm
)	O
.	O
</s>
<s>
Scaling	B-Algorithm
is	O
a	O
linear	B-Architecture
transformation	I-Architecture
,	O
and	O
a	O
special	O
case	O
of	O
homothetic	B-Algorithm
transformation	I-Algorithm
(	O
scaling	B-Algorithm
about	O
a	O
point	O
)	O
.	O
</s>
<s>
In	O
most	O
cases	O
,	O
the	O
homothetic	B-Algorithm
transformations	I-Algorithm
are	O
non-linear	O
transformations	O
.	O
</s>
<s>
A	O
scale	B-Algorithm
factor	I-Algorithm
is	O
usually	O
a	O
decimal	O
which	O
scales	O
,	O
or	O
multiplies	O
,	O
some	O
quantity	O
.	O
</s>
<s>
In	O
the	O
equation	O
y	O
=	O
Cx	O
,	O
C	O
is	O
the	O
scale	B-Algorithm
factor	I-Algorithm
for	O
x	O
.	O
</s>
<s>
For	O
example	O
,	O
doubling	O
distances	O
corresponds	O
to	O
a	O
scale	B-Algorithm
factor	I-Algorithm
of	O
two	O
for	O
distance	O
,	O
while	O
cutting	O
a	O
cake	O
in	O
half	O
results	O
in	O
pieces	O
with	O
a	O
scale	B-Algorithm
factor	I-Algorithm
for	O
volume	O
of	O
one	O
half	O
.	O
</s>
<s>
In	O
the	O
field	O
of	O
measurements	O
,	O
the	O
scale	B-Algorithm
factor	I-Algorithm
of	O
an	O
instrument	O
is	O
sometimes	O
referred	O
to	O
as	O
sensitivity	O
.	O
</s>
<s>
A	O
scaling	B-Algorithm
can	O
be	O
represented	O
by	O
a	O
scaling	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
To	O
scale	O
an	O
object	O
by	O
a	O
vector	O
v	O
=	O
(	O
vx	O
,	O
vy	O
,	O
vz	O
)	O
,	O
each	O
point	O
p	O
=	O
(	O
px	O
,	O
py	O
,	O
pz	O
)	O
would	O
need	O
to	O
be	O
multiplied	O
with	O
this	O
scaling	B-Algorithm
matrix	I-Algorithm
:	O
</s>
<s>
Such	O
a	O
scaling	B-Algorithm
changes	O
the	O
diameter	O
of	O
an	O
object	O
by	O
a	O
factor	O
between	O
the	O
scale	B-Algorithm
factors	I-Algorithm
,	O
the	O
area	O
by	O
a	O
factor	O
between	O
the	O
smallest	O
and	O
the	O
largest	O
product	O
of	O
two	O
scale	B-Algorithm
factors	I-Algorithm
,	O
and	O
the	O
volume	O
by	O
the	O
product	O
of	O
all	O
three	O
.	O
</s>
<s>
The	O
scaling	B-Algorithm
is	O
uniform	O
if	O
and	O
only	O
if	O
the	O
scaling	B-Algorithm
factors	O
are	O
equal	O
(	O
vx	O
=	O
vy	O
=	O
vz	O
)	O
.	O
</s>
<s>
If	O
all	O
except	O
one	O
of	O
the	O
scale	B-Algorithm
factors	I-Algorithm
are	O
equal	O
to	O
1	O
,	O
we	O
have	O
directional	O
scaling	B-Algorithm
.	O
</s>
<s>
In	O
the	O
case	O
where	O
vx	O
=	O
vy	O
=	O
vz	O
=	O
k	O
,	O
scaling	B-Algorithm
increases	O
the	O
area	O
of	O
any	O
surface	O
by	O
a	O
factor	O
of	O
k2	O
and	O
the	O
volume	O
of	O
any	O
solid	O
object	O
by	O
a	O
factor	O
of	O
k3	O
.	O
</s>
<s>
In	O
-dimensional	O
space	O
,	O
uniform	B-Algorithm
scaling	I-Algorithm
by	O
a	O
factor	O
is	O
accomplished	O
by	O
scalar	O
multiplication	O
with	O
,	O
that	O
is	O
,	O
multiplying	O
each	O
coordinate	O
of	O
each	O
point	O
by	O
.	O
</s>
<s>
As	O
a	O
special	O
case	O
of	O
linear	B-Architecture
transformation	I-Architecture
,	O
it	O
can	O
be	O
achieved	O
also	O
by	O
multiplying	O
each	O
point	O
(	O
viewed	O
as	O
a	O
column	O
vector	O
)	O
with	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
whose	O
entries	O
on	O
the	O
diagonal	O
are	O
all	O
equal	O
to	O
,	O
namely	O
.	O
</s>
<s>
Non-uniform	B-Algorithm
scaling	I-Algorithm
is	O
accomplished	O
by	O
multiplication	O
with	O
any	O
symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
eigenvalues	O
of	O
the	O
matrix	B-Architecture
are	O
the	O
scale	B-Algorithm
factors	I-Algorithm
,	O
and	O
the	O
corresponding	O
eigenvectors	O
are	O
the	O
axes	O
along	O
which	O
each	O
scale	B-Algorithm
factor	I-Algorithm
applies	O
.	O
</s>
<s>
A	O
special	O
case	O
is	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
,	O
with	O
arbitrary	O
numbers	O
along	O
the	O
diagonal	O
:	O
the	O
axes	O
of	O
scaling	B-Algorithm
are	O
then	O
the	O
coordinate	O
axes	O
,	O
and	O
the	O
transformation	O
scales	O
along	O
each	O
axis	O
by	O
the	O
factor	O
.	O
</s>
<s>
In	O
uniform	B-Algorithm
scaling	I-Algorithm
with	O
a	O
non-zero	O
scale	B-Algorithm
factor	I-Algorithm
,	O
all	O
non-zero	O
vectors	O
retain	O
their	O
direction	O
(	O
as	O
seen	O
from	O
the	O
origin	O
)	O
,	O
or	O
all	O
have	O
the	O
direction	O
reversed	O
,	O
depending	O
on	O
the	O
sign	O
of	O
the	O
scaling	B-Algorithm
factor	O
.	O
</s>
<s>
In	O
non-uniform	B-Algorithm
scaling	I-Algorithm
only	O
the	O
vectors	O
that	O
belong	O
to	O
an	O
eigenspace	O
will	O
retain	O
their	O
direction	O
.	O
</s>
<s>
To	O
scale	O
an	O
object	O
by	O
a	O
vector	O
v	O
=	O
(	O
vx	O
,	O
vy	O
,	O
vz	O
)	O
,	O
each	O
homogeneous	O
coordinate	O
vector	O
p	O
=	O
(	O
px	O
,	O
py	O
,	O
pz	O
,	O
1	O
)	O
would	O
need	O
to	O
be	O
multiplied	O
with	O
this	O
projective	B-Algorithm
transformation	I-Algorithm
matrix	I-Algorithm
:	O
</s>
<s>
Since	O
the	O
last	O
component	O
of	O
a	O
homogeneous	O
coordinate	O
can	O
be	O
viewed	O
as	O
the	O
denominator	O
of	O
the	O
other	O
three	O
components	O
,	O
a	O
uniform	B-Algorithm
scaling	I-Algorithm
by	O
a	O
common	O
factor	O
s	O
(	O
uniform	B-Algorithm
scaling	I-Algorithm
)	O
can	O
be	O
accomplished	O
by	O
using	O
this	O
scaling	B-Algorithm
matrix	I-Algorithm
:	O
</s>
<s>
If	O
,	O
the	O
transformation	O
is	O
horizontal	O
;	O
when	O
,	O
it	O
is	O
a	O
dilation	B-Algorithm
,	O
when	O
,	O
it	O
is	O
a	O
contraction	O
.	O
</s>
<s>
If	O
,	O
the	O
transformation	O
is	O
vertical	O
;	O
when	O
it	O
is	O
a	O
dilation	B-Algorithm
,	O
when	O
,	O
it	O
is	O
a	O
contraction	O
.	O
</s>
