<s>
In	O
plane	O
geometry	O
,	O
a	O
shear	B-Algorithm
mapping	I-Algorithm
is	O
a	O
linear	B-Architecture
map	I-Architecture
that	O
displaces	O
each	O
point	O
in	O
a	O
fixed	O
direction	O
,	O
by	O
an	O
amount	O
proportional	O
to	O
its	O
signed	O
distance	O
from	O
the	O
line	O
that	O
is	O
parallel	O
to	O
that	O
direction	O
and	O
goes	O
through	O
the	O
origin	O
.	O
</s>
<s>
This	O
type	O
of	O
mapping	O
is	O
also	O
called	O
shear	B-Algorithm
transformation	I-Algorithm
,	O
transvection	B-Algorithm
,	O
or	O
just	O
shearing	O
.	O
</s>
<s>
Shear	B-Algorithm
mappings	I-Algorithm
must	O
not	O
be	O
confused	O
with	O
rotations	O
.	O
</s>
<s>
Applying	O
a	O
shear	B-Algorithm
map	O
to	O
a	O
set	O
of	O
points	O
of	O
the	O
plane	O
will	O
change	O
all	O
angles	O
between	O
them	O
(	O
except	O
straight	O
angles	O
)	O
,	O
and	O
the	O
length	O
of	O
any	O
line	O
segment	O
that	O
is	O
not	O
parallel	O
to	O
the	O
direction	O
of	O
displacement	O
.	O
</s>
<s>
A	O
shear	B-Algorithm
mapping	I-Algorithm
is	O
the	O
main	O
difference	O
between	O
the	O
upright	O
and	O
slanted	O
(	O
or	O
italic	O
)	O
styles	O
of	O
letters	O
.	O
</s>
<s>
This	O
geometric	O
transformation	O
is	O
a	O
linear	B-Architecture
transformation	I-Architecture
of	O
that	O
preserves	O
the	O
-dimensional	O
measure	O
(	O
hypervolume	O
)	O
of	O
any	O
set	O
.	O
</s>
<s>
In	O
the	O
plane	O
,	O
a	O
horizontal	O
shear	B-Algorithm
(	O
or	O
shear	B-Algorithm
parallel	O
to	O
the	O
-axis	O
)	O
is	O
a	O
function	O
that	O
takes	O
a	O
generic	O
point	O
with	O
coordinates	O
to	O
the	O
point	O
;	O
where	O
is	O
a	O
fixed	O
parameter	O
,	O
called	O
the	O
shear	B-Algorithm
factor	O
.	O
</s>
<s>
Vertical	O
lines	O
,	O
in	O
particular	O
,	O
become	O
oblique	O
lines	O
with	O
slope	O
Therefore	O
,	O
the	O
shear	B-Algorithm
factor	O
is	O
the	O
cotangent	O
of	O
the	O
shear	B-Algorithm
angle	O
between	O
the	O
former	O
verticals	O
and	O
the	O
-axis	O
.	O
</s>
<s>
(	O
In	O
the	O
example	O
on	O
the	O
right	O
the	O
square	O
is	O
tilted	O
by	O
30°	O
,	O
so	O
the	O
shear	B-Algorithm
angle	O
is	O
60°	O
.	O
)	O
</s>
<s>
If	O
the	O
coordinates	O
of	O
a	O
point	O
are	O
written	O
as	O
a	O
column	O
vector	O
(	O
a	O
2×1	O
matrix	B-Architecture
)	O
,	O
the	O
shear	B-Algorithm
mapping	I-Algorithm
can	O
be	O
written	O
as	O
multiplication	O
by	O
a	O
2×2	O
matrix	B-Architecture
:	O
</s>
<s>
A	O
vertical	O
shear	B-Algorithm
(	O
or	O
shear	B-Algorithm
parallel	O
to	O
the	O
-axis	O
)	O
of	O
lines	O
is	O
similar	O
,	O
except	O
that	O
the	O
roles	O
of	O
and	O
are	O
swapped	O
.	O
</s>
<s>
It	O
corresponds	O
to	O
multiplying	O
the	O
coordinate	O
vector	O
by	O
the	O
transposed	O
matrix	B-Architecture
:	O
</s>
<s>
The	O
vertical	O
shear	B-Algorithm
displaces	O
points	O
to	O
the	O
right	O
of	O
the	O
-axis	O
up	O
or	O
down	O
,	O
depending	O
on	O
the	O
sign	O
of	O
.	O
</s>
<s>
Horizontal	O
lines	O
,	O
in	O
particular	O
,	O
get	O
tilted	O
by	O
the	O
shear	B-Algorithm
angle	O
to	O
become	O
lines	O
with	O
slope	O
.	O
</s>
<s>
For	O
a	O
vector	O
space	O
and	O
subspace	O
,	O
a	O
shear	B-Algorithm
fixing	O
translates	O
all	O
vectors	O
in	O
a	O
direction	O
parallel	O
to	O
.	O
</s>
<s>
where	O
is	O
a	O
linear	B-Architecture
mapping	I-Architecture
from	O
into	O
.	O
</s>
<s>
The	O
following	O
applications	O
of	O
shear	B-Algorithm
mapping	I-Algorithm
were	O
noted	O
by	O
William	O
Kingdon	O
Clifford	O
:	O
</s>
<s>
"	O
A	O
succession	O
of	O
shears	B-Algorithm
will	O
enable	O
us	O
to	O
reduce	O
any	O
figure	O
bounded	O
by	O
straight	O
lines	O
to	O
a	O
triangle	O
of	O
equal	O
area.	O
"	O
</s>
<s>
"	O
...	O
we	O
may	O
shear	B-Algorithm
any	O
triangle	O
into	O
a	O
right-angled	O
triangle	O
,	O
and	O
this	O
will	O
not	O
alter	O
its	O
area	O
.	O
</s>
<s>
The	O
area-preserving	O
property	O
of	O
a	O
shear	B-Algorithm
mapping	I-Algorithm
can	O
be	O
used	O
for	O
results	O
involving	O
area	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
Pythagorean	O
theorem	O
has	O
been	O
illustrated	O
with	O
shear	B-Algorithm
mapping	I-Algorithm
as	O
well	O
as	O
the	O
related	O
geometric	O
mean	O
theorem	O
.	O
</s>
<s>
An	O
algorithm	O
due	O
to	O
Alan	O
W	O
.	O
Paeth	O
uses	O
a	O
sequence	O
of	O
three	O
shear	B-Algorithm
mappings	I-Algorithm
(	O
horizontal	O
,	O
vertical	O
,	O
then	O
horizontal	O
again	O
)	O
to	O
rotate	O
a	O
digital	B-Algorithm
image	I-Algorithm
by	O
an	O
arbitrary	O
angle	O
.	O
</s>
<s>
The	O
algorithm	O
is	O
very	O
simple	O
to	O
implement	O
,	O
and	O
very	O
efficient	O
,	O
since	O
each	O
step	O
processes	O
only	O
one	O
column	O
or	O
one	O
row	O
of	O
pixels	B-Algorithm
at	O
a	O
time	O
.	O
</s>
<s>
In	O
typography	B-Application
,	O
normal	O
text	O
transformed	O
by	O
a	O
shear	B-Algorithm
mapping	I-Algorithm
results	O
in	O
oblique	O
type	O
.	O
</s>
<s>
In	O
pre-Einsteinian	O
Galilean	O
relativity	O
,	O
transformations	O
between	O
frames	O
of	O
reference	O
are	O
shear	B-Algorithm
mappings	I-Algorithm
called	O
Galilean	O
transformations	O
.	O
</s>
