straightforward question about scale factor
August 08 2008 |
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?
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!
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