<s>
In	O
cartography	O
,	O
a	O
map	B-Application
projection	I-Application
is	O
any	O
of	O
a	O
broad	O
set	O
of	O
transformations	O
employed	O
to	O
represent	O
the	O
curved	O
two-dimensional	O
surface	O
of	O
a	O
globe	O
on	O
a	O
plane	O
.	O
</s>
<s>
In	O
a	O
map	B-Application
projection	I-Application
,	O
coordinates	O
,	O
often	O
expressed	O
as	O
latitude	O
and	O
longitude	O
,	O
of	O
locations	O
from	O
the	O
surface	O
of	O
the	O
globe	O
are	O
transformed	O
to	O
coordinates	O
on	O
a	O
plane	O
.	O
</s>
<s>
Projection	B-Application
is	O
a	O
necessary	O
step	O
in	O
creating	O
a	O
two-dimensional	O
map	O
and	O
is	O
one	O
of	O
the	O
essential	O
elements	O
of	O
cartography	O
.	O
</s>
<s>
All	O
projections	B-Application
of	O
a	O
sphere	O
on	O
a	O
plane	O
necessarily	O
distort	O
the	O
surface	O
in	O
some	O
way	O
and	O
to	O
some	O
extent	O
.	O
</s>
<s>
Depending	O
on	O
the	O
purpose	O
of	O
the	O
map	O
,	O
some	O
distortions	O
are	O
acceptable	O
and	O
others	O
are	O
not	O
;	O
therefore	O
,	O
different	O
map	B-Application
projections	I-Application
exist	O
in	O
order	O
to	O
preserve	O
some	O
properties	O
of	O
the	O
sphere-like	O
body	O
at	O
the	O
expense	O
of	O
other	O
properties	O
.	O
</s>
<s>
The	O
study	O
of	O
map	B-Application
projections	I-Application
is	O
primarily	O
about	O
the	O
characterization	O
of	O
their	O
distortions	O
.	O
</s>
<s>
There	O
is	O
no	O
limit	O
to	O
the	O
number	O
of	O
possible	O
map	B-Application
projections	I-Application
.	O
</s>
<s>
More	O
generally	O
,	O
projections	B-Application
are	O
considered	O
in	O
several	O
fields	O
of	O
pure	O
mathematics	O
,	O
including	O
differential	B-Language
geometry	I-Language
,	O
projective	O
geometry	O
,	O
and	O
manifolds	B-Architecture
.	O
</s>
<s>
However	O
,	O
the	O
term	O
"	O
map	B-Application
projection	I-Application
"	O
refers	O
specifically	O
to	O
a	O
cartographic	B-Application
projection	I-Application
.	O
</s>
<s>
Despite	O
the	O
name	O
's	O
literal	O
meaning	O
,	O
projection	B-Application
is	O
not	O
limited	O
to	O
perspective	B-Architecture
projections	I-Architecture
,	O
such	O
as	O
those	O
resulting	O
from	O
casting	O
a	O
shadow	O
on	O
a	O
screen	O
,	O
or	O
the	O
rectilinear	O
image	O
produced	O
by	O
a	O
pinhole	B-Algorithm
camera	I-Algorithm
on	O
a	O
flat	O
film	O
plate	O
.	O
</s>
<s>
Rather	O
,	O
any	O
mathematical	O
function	O
that	O
transforms	O
coordinates	O
from	O
the	O
curved	O
surface	O
distinctly	O
and	O
smoothly	O
to	O
the	O
plane	O
is	O
a	O
projection	B-Application
.	O
</s>
<s>
Few	O
projections	B-Application
in	O
practical	O
use	O
are	O
perspective	B-Architecture
.	O
</s>
<s>
The	O
Earth	O
and	O
other	O
large	O
celestial	O
bodies	O
are	O
generally	O
better	O
modeled	O
as	O
oblate	O
spheroids	O
,	O
whereas	O
small	O
objects	O
such	O
as	O
asteroids	B-Application
often	O
have	O
irregular	O
shapes	O
.	O
</s>
<s>
Therefore	O
,	O
more	O
generally	O
,	O
a	O
map	B-Application
projection	I-Application
is	O
any	O
method	O
of	O
flattening	O
a	O
continuous	O
curved	O
surface	O
onto	O
a	O
plane	O
.	O
</s>
<s>
The	O
most	O
well-known	O
map	B-Application
projection	I-Application
is	O
the	O
Mercator	O
projection	B-Application
.	O
</s>
<s>
This	O
map	B-Application
projection	I-Application
has	O
the	O
property	O
of	O
being	O
conformal	O
.	O
</s>
<s>
To	O
contrast	O
,	O
equal-area	O
projections	B-Application
such	O
as	O
the	O
Sinusoidal	O
projection	B-Application
and	O
the	O
Gall	O
–	O
Peters	O
projection	B-Application
show	O
the	O
correct	O
sizes	O
of	O
countries	O
relative	O
to	O
each	O
other	O
,	O
but	O
distort	O
angles	O
.	O
</s>
<s>
The	O
National	O
Geographic	O
Society	O
and	O
most	O
atlases	O
favor	O
map	B-Application
projections	I-Application
that	O
compromise	O
between	O
area	O
and	O
angular	O
distortion	O
,	O
such	O
as	O
the	O
Robinson	O
projection	B-Application
and	O
the	O
Winkel	O
tripel	O
projection	B-Application
.	O
</s>
<s>
Map	B-Application
projections	I-Application
can	O
be	O
constructed	O
to	O
preserve	O
some	O
of	O
these	O
properties	O
at	O
the	O
expense	O
of	O
others	O
.	O
</s>
<s>
Similarly	O
,	O
an	O
area-preserving	O
projection	B-Application
can	O
not	O
be	O
conformal	O
,	O
resulting	O
in	O
shapes	O
and	O
bearings	O
distorted	O
in	O
most	O
places	O
of	O
the	O
map	O
.	O
</s>
<s>
Each	O
projection	B-Application
preserves	O
,	O
compromises	O
,	O
or	O
approximates	O
basic	O
metric	O
properties	O
in	O
different	O
ways	O
.	O
</s>
<s>
The	O
purpose	O
of	O
the	O
map	O
determines	O
which	O
projection	B-Application
should	O
form	O
the	O
base	O
for	O
the	O
map	O
.	O
</s>
<s>
Because	O
maps	O
have	O
many	O
different	O
purposes	O
,	O
a	O
diversity	O
of	O
projections	B-Application
have	O
been	O
created	O
to	O
suit	O
those	O
purposes	O
.	O
</s>
<s>
Another	O
consideration	O
in	O
the	O
configuration	O
of	O
a	O
projection	B-Application
is	O
its	O
compatibility	O
with	O
data	O
sets	O
to	O
be	O
used	O
on	O
the	O
map	O
.	O
</s>
<s>
Different	O
datums	O
assign	O
slightly	O
different	O
coordinates	O
to	O
the	O
same	O
location	O
,	O
so	O
in	O
large	O
scale	O
maps	O
,	O
such	O
as	O
those	O
from	O
national	O
mapping	O
systems	O
,	O
it	O
is	O
important	O
to	O
match	O
the	O
datum	O
to	O
the	O
projection	B-Application
.	O
</s>
<s>
Since	O
any	O
map	B-Application
projection	I-Application
is	O
a	O
representation	O
of	O
one	O
of	O
those	O
surfaces	O
on	O
a	O
plane	O
,	O
all	O
map	B-Application
projections	I-Application
distort	O
.	O
</s>
<s>
The	O
classical	O
way	O
of	O
showing	O
the	O
distortion	O
inherent	O
in	O
a	O
projection	B-Application
is	O
to	O
use	O
Tissot	O
's	O
indicatrix	O
.	O
</s>
<s>
Many	O
other	O
ways	O
have	O
been	O
described	O
of	O
showing	O
the	O
distortion	O
in	O
projections	B-Application
.	O
</s>
<s>
In	O
the	O
first	O
half	O
of	O
the	O
20th	O
century	O
,	O
projecting	O
a	O
human	O
head	O
onto	O
different	O
projections	B-Application
was	O
common	O
to	O
show	O
how	O
distortion	O
varies	O
across	O
one	O
projection	B-Application
as	O
compared	O
to	O
another	O
.	O
</s>
<s>
In	O
dynamic	O
media	O
,	O
shapes	O
of	O
familiar	O
coastlines	O
and	O
boundaries	O
can	O
be	O
dragged	O
across	O
an	O
interactive	O
map	O
to	O
show	O
how	O
the	O
projection	B-Application
distorts	O
sizes	O
and	O
shapes	O
according	O
to	O
position	O
on	O
the	O
map	O
.	O
</s>
<s>
The	O
creation	O
of	O
a	O
map	B-Application
projection	I-Application
involves	O
two	O
steps	O
:	O
</s>
<s>
In	O
large-scale	O
maps	O
,	O
Cartesian	O
coordinates	O
normally	O
have	O
a	O
simple	O
relation	O
to	O
eastings	O
and	O
northings	O
defined	O
as	O
a	O
grid	O
superimposed	O
on	O
the	O
projection	B-Application
.	O
</s>
<s>
Some	O
of	O
the	O
simplest	O
map	B-Application
projections	I-Application
are	O
literal	O
projections	B-Application
,	O
as	O
obtained	O
by	O
placing	O
a	O
light	O
source	O
at	O
some	O
definite	O
point	O
relative	O
to	O
the	O
globe	O
and	O
projecting	O
its	O
features	O
onto	O
a	O
specified	O
surface	O
.	O
</s>
<s>
Although	O
most	O
projections	B-Application
are	O
not	O
defined	O
in	O
this	O
way	O
,	O
picturing	O
the	O
light	O
source-globe	O
model	O
can	O
be	O
helpful	O
in	O
understanding	O
the	O
basic	O
concept	O
of	O
a	O
map	B-Application
projection	I-Application
.	O
</s>
<s>
The	O
sphere	O
and	O
ellipsoid	O
do	O
not	O
have	O
developable	O
surfaces	O
,	O
so	O
any	O
projection	B-Application
of	O
them	O
onto	O
a	O
plane	O
will	O
have	O
to	O
distort	O
the	O
image	O
.	O
</s>
<s>
One	O
way	O
of	O
describing	O
a	O
projection	B-Application
is	O
first	O
to	O
project	O
from	O
the	O
Earth	O
's	O
surface	O
to	O
a	O
developable	O
surface	O
such	O
as	O
a	O
cylinder	O
or	O
cone	O
,	O
and	O
then	O
to	O
unroll	O
the	O
surface	O
into	O
a	O
plane	O
.	O
</s>
<s>
If	O
these	O
lines	O
are	O
a	O
parallel	O
of	O
latitude	O
,	O
as	O
in	O
conical	O
projections	B-Application
,	O
it	O
is	O
called	O
a	O
standard	O
parallel	O
.	O
</s>
<s>
The	O
central	O
meridian	O
(	O
usually	O
written	O
λ	O
)	O
and	O
a	O
parallel	O
of	O
origin	O
(	O
usually	O
written	O
φ	O
)	O
are	O
often	O
used	O
to	O
define	O
the	O
origin	O
of	O
the	O
map	B-Application
projection	I-Application
.	O
</s>
<s>
This	O
applies	O
for	O
any	O
cylindrical	O
or	O
pseudocylindrical	O
projection	B-Application
in	O
normal	O
aspect	O
.	O
</s>
<s>
This	O
applies	O
for	O
the	O
Mercator	O
projection	B-Application
in	O
normal	O
aspect	O
.	O
</s>
<s>
This	O
is	O
the	O
defining	O
characteristic	O
of	O
an	O
equidistant	O
projection	B-Application
such	O
as	O
the	O
azimuthal	O
equidistant	O
projection	B-Application
.	O
</s>
<s>
There	O
are	O
also	O
projections	B-Application
(	O
Maurer	O
's	O
two-point	O
equidistant	O
projection	B-Application
,	O
Close	O
)	O
where	O
true	O
distances	O
from	O
two	O
points	O
are	O
preserved	O
.	O
</s>
<s>
Projection	B-Application
construction	O
is	O
also	O
affected	O
by	O
how	O
the	O
shape	O
of	O
the	O
Earth	O
or	O
planetary	O
body	O
is	O
approximated	O
.	O
</s>
<s>
In	O
the	O
following	O
section	O
on	O
projection	B-Application
categories	O
,	O
the	O
earth	O
is	O
taken	O
as	O
a	O
sphere	O
in	O
order	O
to	O
simplify	O
the	O
discussion	O
.	O
</s>
<s>
Therefore	O
,	O
in	O
geoidal	O
projections	B-Application
that	O
preserve	O
such	O
properties	O
,	O
the	O
mapped	O
graticule	O
would	O
deviate	O
from	O
a	O
mapped	O
ellipsoid	O
's	O
graticule	O
.	O
</s>
<s>
Normally	O
the	O
geoid	O
is	O
not	O
used	O
as	O
an	O
Earth	O
model	O
for	O
projections	B-Application
,	O
however	O
,	O
because	O
Earth	O
's	O
shape	O
is	O
very	O
regular	O
,	O
with	O
the	O
undulation	O
of	O
the	O
geoid	O
amounting	O
to	O
less	O
than	O
100m	O
from	O
the	O
ellipsoidal	O
model	O
out	O
of	O
the	O
6.3millionm	O
Earth	O
radius	O
.	O
</s>
<s>
For	O
irregular	O
planetary	O
bodies	O
such	O
as	O
asteroids	B-Application
,	O
however	O
,	O
sometimes	O
models	O
analogous	O
to	O
the	O
geoid	O
are	O
used	O
to	O
project	O
maps	O
from	O
.	O
</s>
<s>
See	O
map	B-Application
projection	I-Application
of	O
the	O
triaxial	O
ellipsoid	O
for	O
further	O
information	O
.	O
</s>
<s>
One	O
way	O
to	O
classify	O
map	B-Application
projections	I-Application
is	O
based	O
on	O
the	O
type	O
of	O
surface	O
onto	O
which	O
the	O
globe	O
is	O
projected	O
.	O
</s>
<s>
In	O
this	O
scheme	O
,	O
the	O
projection	B-Application
process	O
is	O
described	O
as	O
placing	O
a	O
hypothetical	O
projection	B-Application
surface	O
the	O
size	O
of	O
the	O
desired	O
study	O
area	O
in	O
contact	O
with	O
part	O
of	O
the	O
Earth	O
,	O
transferring	O
features	O
of	O
the	O
Earth	O
’s	O
surface	O
onto	O
the	O
projection	B-Application
surface	O
,	O
then	O
unraveling	O
and	O
scaling	O
the	O
projection	B-Application
surface	O
into	O
a	O
flat	O
map	O
.	O
</s>
<s>
The	O
most	O
common	O
projection	B-Application
surfaces	O
are	O
cylindrical	O
(	O
e.g.	O
,	O
Mercator	O
)	O
,	O
conic	O
(	O
e.g.	O
,	O
Albers	O
)	O
,	O
and	O
planar	O
(	O
e.g.	O
,	O
stereographic	O
)	O
.	O
</s>
<s>
Many	O
mathematical	O
projections	B-Application
,	O
however	O
,	O
do	O
not	O
neatly	O
fit	O
into	O
any	O
of	O
these	O
three	O
projection	B-Application
methods	O
.	O
</s>
<s>
Another	O
way	O
to	O
classify	O
projections	B-Application
is	O
according	O
to	O
properties	O
of	O
the	O
model	O
they	O
preserve	O
.	O
</s>
<s>
Because	O
the	O
sphere	O
is	O
not	O
a	O
developable	O
surface	O
,	O
it	O
is	O
impossible	O
to	O
construct	O
a	O
map	B-Application
projection	I-Application
that	O
is	O
both	O
equal-area	O
and	O
conformal	O
.	O
</s>
<s>
The	O
three	O
developable	O
surfaces	O
(	O
plane	O
,	O
cylinder	O
,	O
cone	O
)	O
provide	O
useful	O
models	O
for	O
understanding	O
,	O
describing	O
,	O
and	O
developing	O
map	B-Application
projections	I-Application
.	O
</s>
<s>
For	O
one	O
thing	O
,	O
most	O
world	B-Application
projections	I-Application
in	O
use	O
do	O
not	O
fall	O
into	O
any	O
of	O
those	O
categories	O
.	O
</s>
<s>
For	O
another	O
thing	O
,	O
even	O
most	O
projections	B-Application
that	O
do	O
fall	O
into	O
those	O
categories	O
are	O
not	O
naturally	O
attainable	O
through	O
physical	O
projection	B-Application
.	O
</s>
<s>
Lee	O
's	O
objection	O
refers	O
to	O
the	O
way	O
the	O
terms	O
cylindrical	O
,	O
conic	O
,	O
and	O
planar	O
(	O
azimuthal	O
)	O
have	O
been	O
abstracted	O
in	O
the	O
field	O
of	O
map	B-Application
projections	I-Application
.	O
</s>
<s>
Such	O
a	O
cylindrical	O
projection	B-Application
(	O
for	O
example	O
)	O
is	O
one	O
which	O
:	O
</s>
<s>
Where	O
the	O
light	O
source	O
emanates	O
along	O
the	O
line	O
described	O
in	O
this	O
last	O
constraint	O
is	O
what	O
yields	O
the	O
differences	O
between	O
the	O
various	O
"	O
natural	O
"	O
cylindrical	O
projections	B-Application
.	O
</s>
<s>
But	O
the	O
term	O
cylindrical	O
as	O
used	O
in	O
the	O
field	O
of	O
map	B-Application
projections	I-Application
relaxes	O
the	O
last	O
constraint	O
entirely	O
.	O
</s>
<s>
The	O
famous	O
Mercator	O
projection	B-Application
is	O
one	O
in	O
which	O
the	O
placement	O
of	O
parallels	O
does	O
not	O
arise	O
by	O
projection	B-Application
;	O
instead	O
parallels	O
are	O
placed	O
how	O
they	O
need	O
to	O
be	O
in	O
order	O
to	O
satisfy	O
the	O
property	O
that	O
a	O
course	O
of	O
constant	O
bearing	O
is	O
always	O
plotted	O
as	O
a	O
straight	O
line	O
.	O
</s>
<s>
A	O
normal	O
cylindrical	O
projection	B-Application
is	O
any	O
projection	B-Application
in	O
which	O
meridians	O
are	O
mapped	O
to	O
equally	O
spaced	O
vertical	O
lines	O
and	O
circles	O
of	O
latitude	O
(	O
parallels	O
)	O
are	O
mapped	O
to	O
horizontal	O
lines	O
.	O
</s>
<s>
By	O
the	O
geometry	O
of	O
their	O
construction	O
,	O
cylindrical	O
projections	B-Application
stretch	O
distances	O
east-west	O
.	O
</s>
<s>
The	O
amount	O
of	O
stretch	O
is	O
the	O
same	O
at	O
any	O
chosen	O
latitude	O
on	O
all	O
cylindrical	O
projections	B-Application
,	O
and	O
is	O
given	O
by	O
the	O
secant	O
of	O
the	O
latitude	O
as	O
a	O
multiple	O
of	O
the	O
equator	O
's	O
scale	O
.	O
</s>
<s>
The	O
various	O
cylindrical	O
projections	B-Application
are	O
distinguished	O
from	O
each	O
other	O
solely	O
by	O
their	O
north-south	O
stretching	O
(	O
where	O
latitude	O
is	O
given	O
by	O
φ	O
)	O
:	O
</s>
<s>
North-south	O
stretching	O
grows	O
with	O
latitude	O
faster	O
than	O
east-west	O
stretching	O
(	O
sec	O
φ	O
)	O
:	O
The	O
cylindric	O
perspective	B-Architecture
(	O
or	O
central	O
cylindrical	O
)	O
projection	B-Application
;	O
unsuitable	O
because	O
distortion	O
is	O
even	O
worse	O
than	O
in	O
the	O
Mercator	O
projection	B-Application
.	O
</s>
<s>
North-south	O
stretching	O
grows	O
with	O
latitude	O
,	O
but	O
less	O
quickly	O
than	O
the	O
east-west	O
stretching	O
:	O
such	O
as	O
the	O
Miller	O
cylindrical	O
projection	B-Application
(	O
sec	O
φ	O
)	O
.	O
</s>
<s>
North-south	O
distances	O
neither	O
stretched	O
nor	O
compressed	O
(	O
1	O
)	O
:	O
equirectangular	O
projection	B-Application
or	O
"	O
plate	O
carrée	O
"	O
.	O
</s>
<s>
This	O
projection	B-Application
has	O
many	O
named	O
specializations	O
differing	O
only	O
in	O
the	O
scaling	O
constant	O
,	O
such	O
as	O
the	O
Gall	O
–	O
Peters	O
or	O
Gall	O
orthographic	O
(	O
undistorted	O
at	O
the	O
45°	O
parallels	O
)	O
,	O
Behrmann	O
(	O
undistorted	O
at	O
the	O
30°	O
parallels	O
)	O
,	O
and	O
Lambert	O
cylindrical	O
equal-area	O
(	O
undistorted	O
at	O
the	O
equator	O
)	O
.	O
</s>
<s>
Since	O
this	O
projection	B-Application
scales	O
north-south	O
distances	O
by	O
the	O
reciprocal	O
of	O
east-west	O
stretching	O
,	O
it	O
preserves	O
area	O
at	O
the	O
expense	O
of	O
shapes	O
.	O
</s>
<s>
Normal	O
cylindrical	O
projections	B-Application
map	O
the	O
whole	O
Earth	O
as	O
a	O
finite	O
rectangle	O
,	O
except	O
in	O
the	O
first	O
two	O
cases	O
,	O
where	O
the	O
rectangle	O
stretches	O
infinitely	O
tall	O
while	O
retaining	O
constant	O
width	O
.	O
</s>
<s>
A	O
transverse	O
cylindrical	O
projection	B-Application
is	O
a	O
cylindrical	O
projection	B-Application
that	O
in	O
the	O
tangent	O
case	O
uses	O
a	O
great	O
circle	O
along	O
a	O
meridian	O
as	O
contact	O
line	O
for	O
the	O
cylinder	O
.	O
</s>
<s>
An	O
oblique	O
cylindrical	O
projection	B-Application
aligns	O
with	O
a	O
great	O
circle	O
,	O
but	O
not	O
the	O
equator	O
and	O
not	O
a	O
meridian	O
.	O
</s>
<s>
Pseudocylindrical	O
projections	B-Application
represent	O
the	O
central	O
meridian	O
as	O
a	O
straight	O
line	O
segment	O
.	O
</s>
<s>
Pseudocylindrical	O
projections	B-Application
map	O
parallels	O
as	O
straight	O
lines	O
.	O
</s>
<s>
Examples	O
of	O
pseudocylindrical	O
projections	B-Application
include	O
:	O
</s>
<s>
Sinusoidal	O
,	O
which	O
was	O
the	O
first	O
pseudocylindrical	O
projection	B-Application
developed	O
.	O
</s>
<s>
Collignon	O
projection	B-Application
,	O
which	O
in	O
its	O
most	O
common	O
forms	O
represents	O
each	O
meridian	O
as	O
two	O
straight	O
line	O
segments	O
,	O
one	O
from	O
each	O
pole	O
to	O
the	O
equator	O
.	O
</s>
<s>
The	O
HEALPix	O
projection	B-Application
combines	O
an	O
equal-area	O
cylindrical	O
projection	B-Application
in	O
equatorial	O
regions	O
with	O
the	O
Collignon	O
projection	B-Application
in	O
polar	O
areas	O
.	O
</s>
<s>
The	O
term	O
"	O
conic	O
projection	B-Application
"	O
is	O
used	O
to	O
refer	O
to	O
any	O
projection	B-Application
in	O
which	O
meridians	O
are	O
mapped	O
to	O
equally	O
spaced	O
lines	O
radiating	O
out	O
from	O
the	O
apex	O
and	O
circles	O
of	O
latitude	O
(	O
parallels	O
)	O
are	O
mapped	O
to	O
circular	O
arcs	O
centered	O
on	O
the	O
apex	O
.	O
</s>
<s>
Conic	O
projections	B-Application
that	O
are	O
commonly	O
used	O
are	O
:	O
</s>
<s>
Bonne	O
,	O
an	O
equal-area	O
projection	B-Application
on	O
which	O
most	O
meridians	O
and	O
parallels	O
appear	O
as	O
curved	O
lines	O
.	O
</s>
<s>
American	O
polyconic	O
and	O
other	O
projections	B-Application
in	O
the	O
polyconic	O
projection	B-Application
class	O
.	O
</s>
<s>
Azimuthal	O
projections	B-Application
have	O
the	O
property	O
that	O
directions	O
from	O
a	O
central	O
point	O
are	O
preserved	O
and	O
therefore	O
great	O
circles	O
through	O
the	O
central	O
point	O
are	O
represented	O
by	O
straight	O
lines	O
on	O
the	O
map	O
.	O
</s>
<s>
These	O
projections	B-Application
also	O
have	O
radial	O
symmetry	O
in	O
the	O
scales	O
and	O
hence	O
in	O
the	O
distortions	O
:	O
map	O
distances	O
from	O
the	O
central	O
point	O
are	O
computed	O
by	O
a	O
function	O
r(d )	O
of	O
the	O
true	O
distance	O
d	O
,	O
independent	O
of	O
the	O
angle	O
;	O
correspondingly	O
,	O
circles	O
with	O
the	O
central	O
point	O
as	O
center	O
are	O
mapped	O
into	O
circles	O
which	O
have	O
as	O
center	O
the	O
central	O
point	O
on	O
the	O
map	O
.	O
</s>
<s>
Some	O
azimuthal	O
projections	B-Application
are	O
true	O
perspective	B-Architecture
projections	I-Architecture
;	O
that	O
is	O
,	O
they	O
can	O
be	O
constructed	O
mechanically	O
,	O
projecting	O
the	O
surface	O
of	O
the	O
Earth	O
by	O
extending	O
lines	O
from	O
a	O
point	O
of	O
perspective	B-Architecture
(	O
along	O
an	O
infinite	O
line	O
through	O
the	O
tangent	O
point	O
and	O
the	O
tangent	O
point	O
's	O
antipode	O
)	O
onto	O
the	O
plane	O
:	O
</s>
<s>
The	O
gnomonic	O
projection	B-Application
displays	O
great	O
circles	O
as	O
straight	O
lines	O
.	O
</s>
<s>
Can	O
be	O
constructed	O
by	O
using	O
a	O
point	O
of	O
perspective	B-Architecture
at	O
the	O
center	O
of	O
the	O
Earth	O
.	O
</s>
<s>
The	O
orthographic	O
projection	B-Application
maps	O
each	O
point	O
on	O
the	O
Earth	O
to	O
the	O
closest	O
point	O
on	O
the	O
plane	O
.	O
</s>
<s>
Can	O
be	O
constructed	O
from	O
a	O
point	O
of	O
perspective	B-Architecture
an	O
infinite	O
distance	O
from	O
the	O
tangent	O
point	O
;	O
r(d )	O
=	O
csin	O
.	O
</s>
<s>
Photographs	O
of	O
Earth	O
from	O
far	O
enough	O
away	O
,	O
such	O
as	O
the	O
Moon	O
,	O
approximate	O
this	O
perspective	B-Architecture
.	O
</s>
<s>
Near-sided	O
perspective	B-Architecture
projection	I-Architecture
,	O
which	O
simulates	O
the	O
view	O
from	O
space	O
at	O
a	O
finite	O
distance	O
and	O
therefore	O
shows	O
less	O
than	O
a	O
full	O
hemisphere	O
,	O
such	O
as	O
used	O
in	O
The	O
Blue	O
Marble	O
2012	O
)	O
.	O
</s>
<s>
The	O
General	O
Perspective	B-Architecture
projection	I-Architecture
can	O
be	O
constructed	O
by	O
using	O
a	O
point	O
of	O
perspective	B-Architecture
outside	O
the	O
Earth	O
.	O
</s>
<s>
Photographs	O
of	O
Earth	O
(	O
such	O
as	O
those	O
from	O
the	O
International	O
Space	O
Station	O
)	O
give	O
this	O
perspective	B-Architecture
.	O
</s>
<s>
It	O
is	O
a	O
generalization	O
of	O
near-sided	O
perspective	B-Architecture
projection	I-Architecture
,	O
allowing	O
tilt	O
.	O
</s>
<s>
The	O
stereographic	O
projection	B-Application
,	O
which	O
is	O
conformal	O
,	O
can	O
be	O
constructed	O
by	O
using	O
the	O
tangent	O
point	O
's	O
antipode	O
as	O
the	O
point	O
of	O
perspective	B-Architecture
.	O
</s>
<s>
Other	O
azimuthal	O
projections	B-Application
are	O
not	O
true	O
perspective	B-Architecture
projections	I-Architecture
:	O
</s>
<s>
Polyhedral	O
map	B-Application
projections	I-Application
use	O
a	O
polyhedron	O
to	O
subdivide	O
the	O
globe	O
into	O
faces	O
,	O
and	O
then	O
projects	O
each	O
face	O
to	O
the	O
globe	O
.	O
</s>
<s>
The	O
most	O
well-known	O
polyhedral	O
map	B-Application
projection	I-Application
is	O
Buckminster	O
Fuller	O
's	O
Dymaxion	O
map	O
.	O
</s>
<s>
Conformal	O
,	O
or	O
orthomorphic	O
,	O
map	B-Application
projections	I-Application
preserve	O
angles	O
locally	O
,	O
implying	O
that	O
they	O
map	O
infinitesimal	O
circles	O
of	O
constant	O
size	O
anywhere	O
on	O
the	O
Earth	O
to	O
infinitesimal	O
circles	O
of	O
varying	O
sizes	O
on	O
the	O
map	O
.	O
</s>
<s>
These	O
are	O
some	O
conformal	O
projections	B-Application
:	O
</s>
<s>
These	O
are	O
some	O
projections	B-Application
that	O
preserve	O
area	O
:	O
</s>
<s>
Snyder	O
's	O
equal-area	O
polyhedral	O
projection	B-Application
,	O
used	O
for	O
geodesic	B-Application
grids	I-Application
.	O
</s>
<s>
An	O
equidistant	O
projection	B-Application
preserves	O
distances	O
from	O
one	O
or	O
two	O
special	O
points	O
to	O
all	O
other	O
points	O
.	O
</s>
<s>
Compromise	O
projections	B-Application
give	O
up	O
the	O
idea	O
of	O
perfectly	O
preserving	O
metric	O
properties	O
,	O
seeking	O
instead	O
to	O
strike	O
a	O
balance	O
between	O
distortions	O
,	O
or	O
to	O
simply	O
make	O
things	O
look	O
right	O
.	O
</s>
<s>
Most	O
of	O
these	O
types	O
of	O
projections	B-Application
distort	O
shape	O
in	O
the	O
polar	O
regions	O
more	O
than	O
at	O
the	O
equator	O
.	O
</s>
<s>
These	O
are	O
some	O
compromise	O
projections	B-Application
:	O
</s>
<s>
The	O
mathematics	O
of	O
projection	B-Application
do	O
not	O
permit	O
any	O
particular	O
map	B-Application
projection	I-Application
to	O
be	O
best	O
for	O
everything	O
.	O
</s>
<s>
Thus	O
,	O
many	O
projections	B-Application
exist	O
to	O
serve	O
the	O
many	O
uses	O
of	O
maps	O
and	O
their	O
vast	O
range	O
of	O
scales	O
.	O
</s>
<s>
For	O
smaller-scale	O
maps	O
,	O
such	O
as	O
those	O
spanning	O
continents	O
or	O
the	O
entire	O
world	O
,	O
many	O
projections	B-Application
are	O
in	O
common	O
use	O
according	O
to	O
their	O
fitness	O
for	O
the	O
purpose	O
,	O
such	O
as	O
Winkel	O
tripel	O
,	O
Robinson	O
and	O
Mollweide	O
.	O
</s>
<s>
Reference	O
maps	O
of	O
the	O
world	O
often	O
appear	O
on	O
compromise	O
projections	B-Application
.	O
</s>
<s>
Due	O
to	O
distortions	O
inherent	O
in	O
any	O
map	O
of	O
the	O
world	O
,	O
the	O
choice	O
of	O
projection	B-Application
becomes	O
largely	O
one	O
of	O
aesthetics	O
.	O
</s>
<s>
Thematic	O
maps	O
normally	O
require	O
an	O
equal	O
area	O
projection	B-Application
so	O
that	O
phenomena	O
per	O
unit	O
area	O
are	O
shown	O
in	O
correct	O
proportion	O
.	O
</s>
<s>
The	O
Mercator	O
projection	B-Application
,	O
developed	O
for	O
navigational	O
purposes	O
,	O
has	O
often	O
been	O
used	O
in	O
world	O
maps	O
where	O
other	O
projections	B-Application
would	O
have	O
been	O
more	O
appropriate	O
.	O
</s>
<s>
A	O
controversy	O
in	O
the	O
1980s	O
over	O
the	O
Peters	O
map	O
motivated	O
the	O
American	O
Cartographic	O
Association	O
(	O
now	O
the	O
Cartography	O
and	O
Geographic	O
Information	O
Society	O
)	O
to	O
produce	O
a	O
series	O
of	O
booklets	O
(	O
including	O
Which	O
Map	O
Is	O
Best	O
)	O
designed	O
to	O
educate	O
the	O
public	O
about	O
map	B-Application
projections	I-Application
and	O
distortion	O
in	O
maps	O
.	O
</s>
<s>
In	O
1989	O
and	O
1990	O
,	O
after	O
some	O
internal	O
debate	O
,	O
seven	O
North	O
American	O
geographic	O
organizations	O
adopted	O
a	O
resolution	O
recommending	O
against	O
using	O
any	O
rectangular	O
projection	B-Application
(	O
including	O
Mercator	O
and	O
Gall	O
–	O
Peters	O
)	O
for	O
reference	O
maps	O
of	O
the	O
world	O
.	O
</s>
