Mercator Map Projection (and inverse)

Computes the Mercator map projection, and the inverse projection

This is a very simple function to compute the Mercator projection. You do not need any toolboxes for this to work.

[x,y,scaleFactor] = mercator(lon,lat), -> Mercator Projection
Input:
lon: the longitude of the point or points
lat: the latitude of the point or points
output: x,y values on the mercator projection. To get the values in units
of distance you must multiply this by the radius of the earth and then divide
by the scaleFactor. Note: radius of Earth = 6378.1 kilometers, and each
point will have its own scalefactor, but you should choose only one to
multiply all your points by.

[lon, lat] = mercator(x,y,1), -> Inverse Mercator Projection
(when you enter extra parameter 1, the
inverse is calculated
Input:
x: This must be the 'x' value output by the first call to the
mercator function. It cannot be scaled by anything at this point
if you had scaled it before, you must scale it back.
y: this must be the 'y' value output by the first call to mercator
1: this can be any parameter at all you feel like passing. It will
tell the function that it must compute the inverse mercator.
Output:
lon: the longitude of the point or points
lat: the latitude of the point or points

note that many programs accept or display longitude and latitude in a
different order. I have chosen to use lon,lat in this order so that it
can correspond to the standard for plotting, x,y

Note that you may wish to subtract the mean from the data before plotting
it. The absolute x,y values outputted are entirely meaningless in the
mercator projection. It is important though to add the mean back again if
you would like to compute the inverse.

[x,y,scaleFactor] = mercator(lon,lat);
meanSF = mean(scaleFactor); % use the mean scalefactor (a single number)
x_km=x*6378.1/meanSF; % now x is in units of km
y_km=y*6378.1/meanSF; % now y is in units of km

meanX=mean(x_km);
meanY=mean(y_km);

x_km = x_km-meanX; % the absolute values here are meaningless so you may
y_km = y_km-meanY; % subtract the mean from the data.

plot(x_km,y_km);
axis('equal'); % so that shapes will be preserved (the whole point of this)
xlabel('km'); ylabel('km');