## Saturday, August 28, 2010

### Mercator Scale There are two schools - at least - to plot a line of position with enough details.
Mercator templates are one, plotting sheets are another one.
The plotting sheets rely on the Mercator scale, available at the bottom right corner of the document above. It gives graphically the ratio between one degree of latitude and one degree of longitude.
The Mercator projection relies on the concept of "Increasing Latitude", that gives the value of the latitude, in degrees of longitude...
The increasing latitude λ, for a latitude φ is given by the following formula:

λ(φ) = ln(tan(π/4 + φ/2))

The Mercator scale gives the the ratio between one degree of latitude and one degree of increasing latitude at the given latitude. In consequence, its formula is

ratio = (φ - (φ - 1)) / (λ(φ) - λ(φ - 1))
simplified as
ratio = 1 / (λ(φ) - λ(φ - 1))
also written
ratio = (λ(φ) - λ(φ - 1))-1

That's it!
The main difference I can see between the two methods is that the Mercator templates can be reused, just like a chart, and the plotting sheets are just a draft, to be dropped after used. That's just my opinion. Whoever has a better - or other - one is welcome to speak up.