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    Re: Showing a Mercator Projection is not a perspective projection
    From: Frank Reed
    Date: 2017 Jan 1, 11:47 -0800

    David Pike, you wrote:
    "You’ve fed the mathematical formula for a Mercator projection into a curve sketching programme and overlaid it onto the geometrical projection of a sphere onto a cylinder."

    Except for the "curve sketching programme", yes. It was actually drawn "free-hand" in a drawing program.

    And you added:
    "I hadn’t seen this when I sent my post, because posts seem to be published in batches."

    Today, yes. I broke NavList just before local midnight last night. There's a particular rollover event that occurs once every two to three years. It's the only rollover that's still manual, and I missed one key step last night before I went out. All fixed now. Monthly rollover events are on autopilot until January 1, 2019...

    The math underlying the diagram is simple enough. It's easy geometry to show that a pure perspective cylindrical projection, similar in other respects to the Mercator projection, would have latitude lines spaced in proportion to tan(lat). Or in other words, the meridional parts for that projection would be given by
      m' = 3438·tan(Lat).
    In the original pure spherical Mercator projection (which has been given new life under the rather obnoxious name "Web Mercator"), the spacing between the lines of latitude is proportional to ln[tan(45°+lat/2)], where ln() is the usual calculator lingo for "natural logarithm". Or the meridional parts are
      m = 3438·ln[tan(A)]  where A = 45°+Lat/2.
    Note that in the tropics, for latitudes less than 23.45°, these values differ by less than 3% from the pure perspective projection:
      1 > m / m' > 0.97,
    so for areas near the equator (and also for all areas in the usual Universal Transverse Mercator and nearly all Oblique Transverse Mercator projections) the difference is completely insignificant. It's "OK" to describe the projection as if it's a simple cylindrical projection in those low latitudes --a lamp at the center of a crystal globe projecting outward onto a wrapping cylinder. For latitudes below 41°45', the difference is less than 10%.

    Finally, note that historically there were largely pointless attempts to refine the values of the meridional parts based on the true figure of the Earth. Thus you'll see formulas for the meridional parts that include a small term proportional to sin(Lat) in addition to the primary m term above and other terms beyond that. This doesn't matter in the vast majority of practical applications including practical navigational circumstances. Or to put it differently, the so-called "Web Mercator" projection is just a standard Mercator projection for any practical purpose. This is an on-going source of confusion.

    It's a damn shame --really unfortunate-- that the engineers at Google screwed the world by deciding on a Mercator projection for their mapping standard back around 2005 when they started building Google Maps. They didn't know any better back then. But if you're on a desktop computer, when you click the little "Earth" icon in Google Maps, which displays a "Google Earth" view of the same region, you immediately get a different projection. At large scales (regions smaller than about 100 miles wide), you can't see this, because both projections provide a locally conformal view (identical to a simple plotting sheet!!), guaranteeing that everything "looks right" and shapes on the map are geometrically "similar" to shapes on the ground in the real world. But if you zoom out a bit and then toggle back and forth between the two views, you can see that the projections are different. Google Earth, which was developed later, uses a globe-like perspective projection, showing the region as if viewed from a high altitude. This is much better than the obnoxious Mercator projection, whose only major benefit is straight rhumb lines.

    Frank Reed
    Clockwork Mapping / ReedNavigation.com
    Conanicut Island USA

       
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