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straightforward question about scale factor

August 08 2008 | 4 comments


Given a Mercator projection with a single Latitude of 41°N as origin and a Scale factor at Latitude of Origin = 1.00000.

How do we calculate what the scale factor is at 51°N and 31°N with a Lat of O of 41°N?

Thank you
Alessandro Devic

Mapping Center Answer:

Normally a Mercator projection has the Equator as its Latitude of Origin which of course means that the Scale Factor is 1.0 there (for a tangent projection, which the Mercator projection most often is.) I have heard of people modifying a Mercator projection for a number of reasons. Indeed you could place the latitude of origin at 45 degrees north in which case you would be using what is called an Oblique Mercator projection. However, even then, this case of the projection is usually not constructed as you have asked (see the online web help for the Hotine Oblique Mercator projectionPerhaps if you could tell us your application, we could better advise you about an appropriate projection to use and how to modify it to reduce the distortion in your area of interest. You can also check out the Choosing a Map Projectionsection of the online help

To answer your question, the easiest way to find the scale factor on the Mercator projection is to note that the meridians are equally spaced on the projection, yet converge on the globe according to the cosine of the latitude. Hence the east-west distortion must be 1.0/cos(lat) or sec(lat). Since the projection is conformal, the north-south distortion is the same as the east-west at any point and hence is sec(lat).

Calculating scale factor at one point on a map is pretty straight forward.  Calculating scale factor across the wholemap can be a challenge. There are some software packages like GeoCartthat include distortion diagrams. Here is an article that discusses calcuating scale factor. Essentially you need to know the Actual Scale and Principal Scale at all points to be able to do this.

Hope this helps!

pipeline projection posted by Alessandro Devic on Nov 27 2014 10:00PM
thank you, it was helpful. I'd like to re-project these two pipeline maps of Europe and Russia, but I cant find any projection information written on either map. Just by looking at them would you be able to make some educated guesses as to which projections they might be in?
That would be great - there are so many projections, I'm a bit lost trying to figure out which ones they are. ... NE2014.pdf ... h_id=26386

Alessandro Devic
ps posted by Alessandro Devic on Nov 27 2014 10:05PM
What is your ide about Van der Grinten IV: finding the longitude was trivial, but the latitude eludes me. Do you know of a closed-form inverse for the latitude? I have a feeling there is no such thing, as the formula for latitude is fairly complicated.

P.S. I'm pretty sure both the American polyconic and the Rectangular polyconic have no closed-form inverse. But I would love to be corrected.
To find the projection... posted by Aileen Buckley on Dec 11 2014 3:31PM
To find the projection (or coordinate system) when it is not known, try the workflow suggested in the Coordinate System Decision Tree-- you can find the link to this on the Mapping Center > ArcGIS Resources > More page ( The decision tree is the second item in the table. Hope this helps!
Van der Grinten iV posted by Aileen Buckley on Dec 11 2014 4:03PM
We don't support Van der Grinten IV (just I), so I'm not sure about that question. Looking at the forward equations in Snyder and Voxland, An Album of Map Projections, I’m pretty sure there isn’t a closed form.

Our implementation of the American polyconic uses a Newton-Raphson method for the inverse.

A possibility is to ask your question on CartoTalk,, or possibly the GIS stackexchange,

Actually, it’s probably a better idea to ask on CartoTalk first. daan Strebe often can figure these things out (he’s on CartoTalk).

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