[Proj] How to Scale a Mercator projection?

[Christopher Barker]

Gerald I. Evenden wrote:
> On Tuesday 25 July 2006 5:58 pm, Christopher Barker wrote:
>>ValueError: Failed to initialize PROJ.4 with `+proj=merc +lat_ts=29.1333
>>+lon_0=-89.583333 +k=0.000000 +x_0=100 +y_0=-2972000 +ellps=WGS84
>>+datum=WGS84 +units=m +no_defs '

> The above should generate an error:

And indeed it does. Actually, that error was helpful, because when I got 
it, I could see what parameters were being passed into proj by the 
wrapper code.

> In the case of proj=merc, lat_ts= when specified overides any specification of 
> k_0.

I've figured that out now.

> lat_ts is commonly used on navigation charts.  If k_0 is used, I suspect it is 
> only used in trans-equator maps.

Well, I'm still confused. What does lat_ts mean? There is this 
unfortunate typo in the docs:

"the +lat_ts parameter, which is the latitude at which the scale is 0."

Clearly that isn't supposed to be zero, but what is it supposed to be?

On the NOAA Navigational charts, a latitude is given at which the given 
scale holds. That sounds to me like the latitude of true scale, but in 
that case, you'd need the k_0 paramter, in addition to lat_ts.

How do I match the projection of a nautical chart?

Frank Warmerdam wrote:
> My understanding is that PROJ.4 does not support a latitude of origin
> (ie. lat_0) other than the equator.

I did see that in the docs, but I'm not sure that a latitude of origin 
is the same as the latitude at which the given scale is correct. If it 
it is, then perhaps it's time to patch PROJ to allow that - Nautical 
charts are the most common use of the mercator projection, and the math 
is just not that hard for this one.

I found this in the FAQ about the NOAA arc-view extension for reading 
Nautical charts:

************
4.3 What's the "Latitude of True Scale" that Mercator charts ask for?

The latitude of true scale in a Mercator projection is the one latitude 
for which distance measurements are correct. In general, a Mercator 
chart will have a latitude of true scale in the center of the chart. 
However, if the latitude of true scale given in the chart were applied, 
it would not be possible to put multiple Mercator charts in a single 
view. Therefore the developers have provided a means for allowing the 
user to set the latitude of true scale that will be used for all 
Mercator charts in a given view. When setting the latitude of true scale 
for a view, two options are suggested: 0.0 degrees and the latitude of 
true scale that is read from the chart being added. The drawback to 
using 0.0 degrees is the resulting inaccurate measurements. Few, if any, 
NOAA nautical charts actually have the equator in them; thus, the scale 
is wrong everywhere in the chart and the error increases with latitude.
To allow the user to get reasonable distance measurements, a latitude of 
true scale may be entered for the first Mercator chart added to the 
view. Additional charts will use that latitude of true scale, so it is 
wise to pick a value in the center of your anticipated work area. This 
value does not have to be the original latitude of true scale of the 
chart; you may want to pick a value that is close to the center of your 
study area. Note that there will always be distortion, and therefore 
error in distance measurements not on the latitude of true scale.
************

This implies to me that you should be able to set lat_ts and k_0. A 
latitude of true scale without a scale isn't very useful. I guess you'd 
get a scale of 1 there, which I guess means that units would be in 
meters. Then I'd need to do the pixels-to-meters calculation after the 
fact. Not a big deal, but I was hoping that it wouldn't be neccesary.

As a note, it seems that the ArcView extension isn't really doing a real 
projection anyway -- all they do is determine the location of the corner 
  and the average size of the pixels from the reference points. Maybe I 
should just do that.

Thanks for your help,

-Chris