A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2016 Dec 26, 12:31 -0800
Gary LaPook, you wrote:
"I understand that for absolute accuracy the small scale plotting sheet should not be called "Mercator" but then why, in all these years, has there not been a different name for it put forward in any of the navigation texts?"
Sorry... I should have said so. There is indeed another name commonly used. They are often called "universal" plotting sheets, implying, we may suppose that they can be used more or less anywhere on the globe, which is true (in practice, it would be a mistake to use them in very high latitudes, and in fact, most of these have longitude-scaling curves that limit them to latitudes below 70°). And "universal" is an adequate name, though it doesn't really explain much. Universal is "OK", but it is wrong to call these plotting sheets Mercator plotting sheets, just as it would be wrong to call them Lambert plotting sheets. Let's be clear, both the Mercator projection and the Lambert Conformal projection are "locally conformal" and therefore a standard longitude-scaled plotting sheet, sometimes known as a "universal plotting sheet" is a "postage stamp" slice of a tiny area from either of those projections --or any other conformal map projection.
You also wrote:
"You propose several names that I have never seen before in any text."
Well now that just isn't true. I used the word "conformal" which you have most certainly seen many times before. The Lambert Conformal Projection is quite common in aviation charts. If you visit the Wikipedia article on this projection, you will find that an aeronautical chart is displayed as the primary example. So no doubt you have seen the word "conformal" in a mapping context thousands of times in your life. But perhaps it didn't stick! I also used the word "similar", but I should be clear that I was using that name in an effort to describe the meaning of conformal since "similar triangles" is a mathematical concept that every child learns at some point, probably many times. The idea of similar triangles is basically identical to conformality in map projections.
"Bowditch says that you can make a Mercator small area plotting sheet by using Meridional parts and then goes on to say "a good approximation can be more quickly constructed" using the described methods. "A good approximation" of what, of a Mercator chart."
Ha ha ha haaaa... Oh, good lord, if it wasn't so painful it would be goddam hilarious. What you have described here is the Bowditch directions for making a homemade Mercator chart of a region of the world covering some several degrees --not a standard plotting chart! That's what meridional parts are required for. The meridional parts are indeed the defining mathematical element of the Mercator projection. The values in a table of meridional parts are calculated by integrating the simple longitude-scaling factor which is the basis for standard small plotting charts (the ones sometimes called "universal plotting sheets"). You take the basic relationship: dLon = miles/cos(lat) and if you integrate that, adding up from one small step in latitude to another, you get the meridional parts (*see PS) relation m = 7916·log[tan(45°+lat/2)]. In that section in Bowditch, they reference meridional parts because you need those values in order to produce a genuine Mercator projection of the spherical Earth (*see PS again). You sure as hell don't need those numbers to produce a simple plotting chart! For a conformal chart, meaning a chart that preserves angles so the azimuths of the intercept plot are correct, you only need longitude-scaling, usually taken from a set of curves at the bottom of the chart.
Something else to consider... A standard plotting sheet has a fixed scale of latitude on the central meridian. We scale the miles of longitude to match by reducing those miles by the factor cos(lat) to get the correct dLon values (or vice versa). Note that this is exactly opposite the relationship found on standard small plotting sheets. On "universal plotting sheets" the separation between the lines of latitude is fixed, and the spacing of longitude lines is reduced. This, by the way, is why you can't make a Mercator chart by patching together multiple plotting sheets for different latitudes. You can do so if you use blank sheets of paper, which then allow you to vary the latitude scale, but standard small plotting charts just won't do. That's yet more evidence that they are not Mercator charts --you can't "patch together" these small charts to produce a large Mercator chart.
To make a realistic-looking map (meaning conformal, shape-preserving) of any part of the world smaller than about 200 miles on a side in regions below about 75° latitude, all you need to do is multiply longitude differences by cos(lat), or equivalently you divide east-west distances in miles by cos(lat) to get differences in longitude. There is no need for meridional parts. There is no need for a Mercator projection. Indeed there is no need for any further mapping projection information at all except the basic orientation info (minutes of longitude increase to the right in the eastern hemisphere, to the left in the western hemisphere, in the usual plotting convention). And guess what: that is exactly what common plotting sheets provide. They're mislabeled as "Mercator projections" because some unfortunate idiot made that choice decades ago when preparing a print order. The original idiot probably went to his boss, well aware they weren't Mercator charts, and asked, "Hey, how should we label these things?" And that boss probably snapped at him, "Don't you know? That is obviously a Mercator chart!" Said idiot then did what he was told and kept his job. And yet more idiots copied that idiot in the decades that followed. This is how entropy erodes a technical field.
"I think you may be being overly pedantic on this one, Frank."
I'll convince you yet! Or if not you personally, then someone else who has had the same attitude.
Conanicut Island USA
PS: The spherical calculation of meridional parts results in the short equation 7916·logtan(A) where A is 45°+lat/2, but there are often progressively smaller corrections added to this. For example, standard meridional parts tables usually include a term of -23·sin(lat), if I remember correctly. These additional corrections are not especially significant to navigation, but for some reason they became an object of obsession at various points in the history of navigation.