<s>
In	O
differential	B-Language
geometry	I-Language
,	O
an	O
equiareal	B-Algorithm
map	I-Algorithm
,	O
sometimes	O
called	O
an	O
authalic	O
map	O
,	O
is	O
a	O
smooth	O
map	O
from	O
one	O
surface	O
to	O
another	O
that	O
preserves	O
the	O
areas	O
of	O
figures	O
.	O
</s>
<s>
If	O
M	O
and	O
N	O
are	O
two	O
Riemannian	B-Architecture
(	O
or	O
pseudo-Riemannian	O
)	O
surfaces	O
,	O
then	O
an	O
equiareal	B-Algorithm
map	I-Algorithm
f	O
from	O
M	O
to	O
N	O
can	O
be	O
characterized	O
by	O
any	O
of	O
the	O
following	O
equivalent	O
conditions	O
:	O
</s>
<s>
An	O
example	O
of	O
an	O
equiareal	B-Algorithm
map	I-Algorithm
,	O
due	O
to	O
Archimedes	O
of	O
Syracuse	O
,	O
is	O
the	O
projection	B-Application
from	O
the	O
unit	O
sphere	O
to	O
the	O
unit	O
cylinder	O
outward	O
from	O
their	O
common	O
axis	O
.	O
</s>
<s>
Every	O
Euclidean	B-Algorithm
isometry	I-Algorithm
of	O
the	O
Euclidean	O
plane	O
is	O
equiareal	B-Algorithm
,	O
but	O
the	O
converse	O
is	O
not	O
true	O
.	O
</s>
<s>
In	O
fact	O
,	O
shear	B-Algorithm
mapping	I-Algorithm
and	O
squeeze	O
mapping	O
are	O
counterexamples	O
to	O
the	O
converse	O
.	O
</s>
<s>
Shear	B-Algorithm
mapping	I-Algorithm
takes	O
a	O
rectangle	O
to	O
a	O
parallelogram	O
of	O
the	O
same	O
area	O
.	O
</s>
<s>
Gaussian	B-Algorithm
elimination	I-Algorithm
shows	O
that	O
every	O
equiareal	B-Algorithm
linear	O
transformation	O
(	O
rotations	B-General_Concept
included	O
)	O
can	O
be	O
obtained	O
by	O
composing	O
at	O
most	O
two	O
shears	B-Algorithm
along	O
the	O
axes	O
,	O
a	O
squeeze	O
and	O
(	O
if	O
the	O
determinant	O
is	O
negative	O
)	O
,	O
a	O
reflection	B-Algorithm
.	O
</s>
<s>
In	O
the	O
context	O
of	O
geographic	O
maps	O
,	O
a	O
map	B-Application
projection	I-Application
is	O
called	O
equal-area	O
,	O
equivalent	O
,	O
authalic	O
,	O
equiareal	B-Algorithm
,	O
or	O
area-preserving	B-Algorithm
,	O
if	O
areas	O
are	O
preserved	O
up	O
to	O
a	O
constant	O
factor	O
;	O
embedding	O
the	O
target	O
map	O
,	O
usually	O
considered	O
a	O
subset	O
of	O
R2	O
,	O
in	O
the	O
obvious	O
way	O
in	O
R3	O
,	O
the	O
requirement	O
above	O
then	O
is	O
weakened	O
to	O
:	O
</s>
<s>
For	O
examples	O
of	O
such	O
projections	B-Application
,	O
see	O
equal-area	O
map	B-Application
projection	I-Application
.	O
</s>
