A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Sean C
Date: 2016 Dec 31, 02:06 -0800
I would argue that the neither of the two graphical methods described in Bowditch would produce an approximation of a Mercator projection. They are really just two slightly different ways of doing the same thing: varying the relative spacing and scale of the meridians. There are two very slight differences between them. One: with the first method, one starts with the meridian(s) and then draws the parallels. The second method is, of course, vice versa. But, the end result is exactly the same. No matter which method one chooses, one ends up with the same relative spacing and scale. This is because the relative spacing and scale in both methods is determined by the length of the oblique line drawn at an angle (always with a parallel) that is equal to the latitude. And two: the absolute scale of the sheet is more conveniently determined using the second method. This is because, when starting with the parallel(s), one determines the scale of nautical miles to inches or centimeters directly and derives the spacing of meridians from that. Whereas, when starting with the meridian(s), the length of the oblique line (and thus, the scale of nautical miles to inches or centimeters) is determined by the [likely arbitrary] spacing of the meridians and the angle of the latitude.
If one were to try and produce a world map (or even a very small scale chart) with either of these methods, the result would be an increasing distortion, in the horizontal axis only, proportional to the increase in latitude. The parallels, just as on a globe, would all be equally spaced. (Ignoring, of course, that Earth is oblate.) Area, even in very small sections, would be highly distorted the further away one gets from the mid-latitude. It would probably resemble an equirectangular projection map. This is why these types of plotting sheets are not recommended for use in very high latitudes: because the rate of increase of distortion becomes too great. The Mercator projection, by its nature, distorts both the horizontal and vertical axes. This preserves area in small sections, but distorts the relative area over large sections. This is where meridional parts become useful. Defined [in Bowditch] as "the length of the arc of a meridian between the equator and a given parallel on a Mercator chart..." they allow one to vary the spacing of parallels over very small areas, thus distorting the vertical in proportion to the horizontal distortion which results from keeping meridians equidistant along their length. This allows for less distortion (or rather, more equal distortion) of small areas at high latitudes. And this is where the difference between a Mercator chart and a simple conformal plotting sheet would become obvious, even without a microscope. ;)
Perhaps one could construct a conformal plotting sheet for high latitudes by using different oblique lines for each parallel and drawing the meridians in as curves. But, I've never tried it and I'm not sure it would be convenient or produce an accurate enough (or even useful) approximation for plotting.
Now, admitting that I only have a high school education, barely made it through algebra and cannot claim anything beyond a neophyte's understanding of the topics discussed on NavList: I'll gladly accept any corrections to, or criticism of, the above.