A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2016 Dec 26, 08:56 -0800
David Pike, you wrote:
"I agree the universal plotting chartlet described by you is not Mercator straight off the pad, but when you divide the nm easting by the cosine of the latitude to get chlong, aren’t you effectively doing something very similar to what Mercator incorporated automatically into his chart. That’s why it’s stood the test of time. "
Time to scream. Arrrrrgh. No, that is exactly the big, fat error that navigators routinely make. Local scaling of longitude minutes to match latitude minutes is not the Mercator projection. That property describes a map that is locally conformal, but various map projections are locally conformal, not just Mercator. If you don't like the name "conformal," which is unfortunately not very transparent in meaning, then call it "longitude-scaled." Another name for it would be "similarity". On a locally conformal map projection squares look like squares, and triangles look like triangles. That is, the shapes portrayed on the map are "similar," in the sense of "similar triangles," to shapes drawn on the ground, in the real world. Note that this only applies locally, in a small area, on a Mercator chart. The best example of a failure to preserve shapes in the global Mercator chart is the rhumbline itself which is a spiral, a loxodrome, on the globe, but a perfectly straight line on the chart. In each very small local region shapes are portrayed correctly, but globally they cannot be.
The Mercator projection employs the simplest form of integration (in the calculus sense) of local conformality to produce a global projection. Indeed, the mathematical equation which defines the Mercator projections y-values, also known as meridional parts, is nothing more than the integration of the differential relationship dm = dlat / cos(lat). That is, if you take the local scaling factor, cos(lat), which provides local conformality, and then extend it from one central latitude to the next, advancing steadily up the chart in latitude in arbitrarily tiny steps, the cumulative values of the y-coordinates on the chart are the integral of 1/cos(lat) which is, among other equivalent forms, log[tan(45°+lat/2)]. This defines the global Mercator projection.
Imagine taking a postage stamp-sized bit from a wall-sized Mercator chart. That's like a differential element of the intregrated Mercator projection. This will necessarily be locally conformal. In fact, that's the only aspect of the Mercator projection that remains when you take a tiny piece of it. It's also the only property that is required of a local "universal plotting sheet". Can't we then say that a plotting chart is just a little piece of a Mercator chart? Yes, but it is also a little piece --a postage stamp-sized slice-- pulled from any conformal chart. The small plotting charts used for plots of the intercept method are merely locally conformal. They are not identical to or equivalent to Mercator charts anymore than they are identical or equivalent to Lambert conformal charts (frequently used in aviation), which are, as the name suggests, also conformal. Longitude-scaling is a basic feature of many mapping projections. It's not unique to the Mercator projection, and it is not correct to call these little plotting sheets Mercator charts.
But come on, isn't it close enough? And doesn't frequent usage win out in the end? Sure. In most of the English language, usage is king. Usage defines words, not dictionaries. But in technical and scientific fields, this attitude can be dangerous, and it is generally avoided. In science and engineering, words have relatively fixed meanings, and shifting usage is counted as entropy, flaw, laziness, and just plain wrong usage. Generally speaking, navigators abide by this scientific attitude towards meaning and even tend to be obsessive about exact meanings of words. Is it "OK" to say that the Sun reaches its "zenith" when it has climbed to its maximum altitude on the meridian?? That's a common expression in normal English usage. But in navigation, the zenith is one point in the horizontal coordinate system: the point located 90° above the true horizon. Shifting usage is counted as incorrect usage. Surely then we must apply the same standard to mapping and charting terminology. Despite the fact that many small plotting sheets have been labeled Mercator for decades, this must be counted as erroneous. They are not Mercator charts. They are simple conformal or "longitude-scaled" plots. Usage is not king in a technical, scientific subject like navigation. Fixed, correct definitions should be respected. And finally I would add that navigators are the only users of longitude-scaled plots who commonly make this mistake. It's not a general error of all users of Mercator projections and the broader category of conformal projections: it's a specific error in the culture of modern navigation.
Conanicut Island USA