Mercator Projection
Why Mercator ?
There are numerous possible map projections for depicting the Earth's surface on a map that
differ in use and appearance.
Criteria for selecting a suitable projection include:
-
Appearance - simplest evaluation by optic comparison of projections.
-
Distortion characteristics - determined by the type of the projection. Here there is a
distinction between distances, planes and angle distortion.
-
Suitability for distances and calculation of linear distances.
One of the most important requirements for our purposes is the conformality (angle accuracy)
of the representation.
The advantages of conformality are:
-
Small sections of the Earth's surface are presented without distortion. Conformal projection
gives you a familiar view of your area.
-
It is not necessary to take varying vertical distortions in different directions into
consideration when searching for the next angle in routing.
The requirement for conformality, together with the demand to depict the world "in which we
live" leads to the most well known map representation, the Mercator Projection. For these
reasons, the Mercator Projection is also the most commonly used representation used in
navigation at sea.
One of the most significant characteristics of this projection is a pure latitude-dependent
vertical stretching. Equatorial areas are undistorted; as latitude increases, the distortion of the
Earth's surface grows larger. This leads to an oversized representation towards the poles at
high latitudes and in areas stretched in a North/South direction such as Greenland or
Scandinavia (see map).
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Equation for Representation:
π
x = R ⋅ 180
⋅ L + Off x
L = ( x − Off x ) ⋅ 180
⋅ π1
R
π
y = R ⋅ Ln(Tan( π4 + 360
⋅ B )) + Off y
B=
with:
360
π
⋅ ( ArcTan(Exp( (y- Off y ) / R )) – π4 )
R = 6371000m
-
Longitude (decimal)
-
Latitude (decimal)
-
Offset in x-direction (Null)
-
Offset in y-direction (Null)
The choice of suitable offset is still open. There are 3 possibilities in general:
1. No offset - the origin of the map projection remains in the origin of the geographic coordinates. The only disadvantage is with negative co-ordinates.
Off = Off y = 0 ,
Important: This offset is also chosen for the sake of simplicity. No disadvantages result.
The formulae above are simplified as a result.
2. Move the origin of the Mercator Projection approximately in the origin of the superconformal
projection (south-west of the Azores). Advantage: all co-ordinates in Europe fit in 3 bytes.
Off x = 3400000 , Off y = −1000000
3. Select the offset in such a way that the co-ordinates are always positive. In this case, it
must be considered how best to divide the Earth in the North/South direction in such a
world projection.
Off x = 22000000 , Off y = 16000000
Correction: The offY must be set significantly higher, e.g., to half the value range of the
variables ( 2^31 ? ).
It must be taken into account how island groups, e.g., Hawaii or the Aleutians on the
dividing line of the map are handled (see next point).
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Problems with World Projection
With world projections it must not be forgotten that the Earth must be divided up somewhere
when presented on a map. It is probably a good idea to take the political affiliation of the islands
(groups) on the dividing line into account.
Other problems could arise if linear distances are calculated or routed beyond these borders.
Route Calculation with Map Co-Ordinates
For the calculation of distances (e.g., linear distances) from map co-ordinates distortion by a
latitude-dependent factor must be taken into consideration.
s = Cos( B ) ⋅ ∆x 2 + ∆y 2
The factor takes the vertical stretching in the Mercator Projection into account. This
approximation formula is sufficiently accurate for our needs for distances of up to 600 km and
80° latitude (the error is never more than 5% even for extreme values).
Converting Real Distance to "Mercator Distance"
The trigonometric functions must expect the radiant as argument (should really be standard).
Here is an example in Perl because certain functions don't belong to the basic configuration in
Perl (atan()).
$pi=atan2(1,1)*4;
pi=3.141...
R=6371000;
# (Perl)
# Earth radius [m]
Approximation co-ordinate (Mercator): x,y
e.g., from the start point, y is necessary.
sreal - distance in metres
smerc - distance in Mercator co-ordinates:
$smerc = $sreal/cos(2*(atan2(exp($y/$R),1)-$pi/4));
# Perl
smerc = sreal/cos(2*(atan (exp( y/R) )- pi/4))
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Transformations in Perl:
$pi=atan2(1,1)*4; # Pi
$rho=180/$pi;
# Special constants for geodata
$R=6371000;
# Earth radius [m]
sub geo2merc()
# Conversion Geographical -> Mercator
{
# Call: (x,y)=geo2merc( $Longitude [deg*100000], $Latitude [DEG*100000]);
my( $x,$y,$l,$b,$h);
$l=$_[0]/100000;
# Within PTV geographical co-ordinates
$b=$_[1]/100000; # represented as integer with displaced decimal point.
$x=int(0.5+$l/$rho*$R);
$h=($b/(2*$rho)+$pi/4);
$y=int(0.5+$R*log(sin($h)/cos($h))); # sin()/cos()= tan(), is not defined in Perl !
return($x,$y);
}
sub merc2geo() # Call: (longitude,latitude)=merc2geo(x,y)
{
my( $x, $y, $l, $b );
$x=$_[0];
$y=$_[1];
print " \$x, \$y = $x, $y\n";
$l= $x*$rho/$R;
$b= 2*$rho*(atan2(exp($y/$R),1)-$pi/4);
print " \$l, \$b = $l, $b\n";
return($l*100000,$b*100000);
}
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Europe in Mercator Projection
In this Mercator Projection you can see clearly from the example of Scandinavia how areas at
high latitudes are enlarged.
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For comparison Europe in the Superconformal Projection
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The world in Mercator Projection
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