[vajvywgs] Mercator projection distortion in the United States

linear scale factor of the Mercator projection is sec(latitude).  area scale factor is the square of that.

U.S. lower 48 states extends from about 25 degrees north latitude (south Florida, south Texas), 30 degrees (Gulf coast) to 49 degrees north (straight-ish border with Canada across the west and midwest).

sec(25 deg) = 1.103
sec(30 deg) = 1.1547
sec(49 deg) = 1.524

sec(49 deg)/sec(25 deg) = 1.381

square of that = 1.908

lengths in northern states are "too long" by almost 40%; areas are too large by almost double.

Mercator as Web Mercator is very common.  inspired by national weather maps: storm systems in the north look bigger.  perhaps they therefore earn more disaster relief: spoils from winning the Civil War.

in its defense, Mercator is a good projection for conveying raw data: it is easy to determine the latitude and longitude of any given projected point, so easy to re-project data to another map projection.  (getting latitude involves arctan exp, also called the Gudermannian.)  Mercator is also conformal, and that might be scientifically important: is a hurricane eye a circle or an ellipse?  there are likely important meteorological implications of a non-circular eye.  there is no map projection that is both conformal and equal area.

what country suffers the most Mercator distortion between its northern and southern regions?  candidates: Russia, Chile, Greenland, United States (including Alaska and Hawaii), Japan.

how does the Mercator area of Alaska compare to the rest of the United States?

draw a different Mercator projection, a transverse Mercator projection, that makes the South too large than the North by the same factor (1.381) that normal Mercator makes the North too large.  what great circle is the "equator" of this transverse Mercator?  the equator suffers the least area distortion.

draw another transverse Mercator projection with the "equator" running through the middle of the U.S lower 48, trying to minimize area distortion while maintaining conformality.  exactly which great circle should the "equator" be?  for a given region on a sphere, what conformal map projection minimizes the maximum area distortion?

it is not easy to recover latitude and longitude from a traverse Mercator projection.

we now discuss aspect ratio in normal Mercator projection.  the U.S. lower 48 spans about 24 degrees of latitude as described above.  Maine is at longitude -67 (67 west) and northwest Washington state is at -125, so a span of about 58 degrees of longitude.  its aspect ratio in normal Mercator projection is 1.9, wider than tall.  calculating height in Mercator projection involves tangent and logarithm.  (the vertical asymptote of the tangent function causes height to diverge to infinity at the poles.  the input to the logarithm function is always a number greater than one, so logarithm's vertical asymptote at zero never comes into play.)