[UPDATED: I've redone the images, so they should be visible this time. Sorry.]
Former Alaskan and aviation faculty member and a pilot, Bill Butler, offered some followup info and images to my
Earth Null School post. I asked if I could post his email as a guest post. He graciously agreed. So here's a little more on this topic, which because it deals very visually with distortions of "how we know," is very dear to this blog's underlying theme.
I'm told, that geography isn't taught much in school these days. My sense is that there are people who really, really know this stuff, people like me who sort of know it, and then most other people who don't have a clue.
With GPS people don't even need to read maps at all any more. But understanding this helps us understand other geo-political issues. Plus maps are a good metaphor for other representations of reality that aren't as tangible - like words and theories. And this issue about Mercator projections and how they distort size is certainly important to Alaskans. (Relax, even after undistorting us, we're still bigger than Texas.)
So, for those of you for whom this is a stretch, I invite you to do a little mental yoga. I'm also trying to use parts of my brain that usually don't have to move at all. And so, you'll see at the end of this post, my questions to Bill, and his further response.
Part I - Bill's Guest Post
Because Mercator "straightens" the meridians of longitude, making them
appear parallel instead of converging at the poles, he needed to
mathematically stretch the distance between the parallels of latitude to
keep proper relations between two points in the
same region and then slice open the resulting cylinder to make a flat
surface. Note how the squares (known as graticules) become rectangular
toward the poles. The distortions are huge and obvious when we hang a
map of the world on the classroom wall, but
if our object is to make an accurate map for a voyage from, say--Norway
to Scotland--they are insignificant :
[Steve's note: Greenland (green) is mostly between 0˚ and 60˚ W and between 90˚ and 60˚N. Brazil (red) is between 30˚ and 60˚W and 0˚ and 30˚ S]
On Mercator's Projection, the only place where there is no distortion is
along the equator. The farther you move from it, the greater the
distortion. The traditional comparison is Greenland to Brazil.
Brazil
is 2.5 times the size of Greenland, but if you
are navigating from Thule to Godthaab, that doesn't matter.
Johann Lambert (1728-1777, an Alsatian) figured out that if instead of
creating Mercator's cylinder, you built a cone, it could also touch the
surface at some parallel other than the equator and that would be the
circle of zero distortion, and by moving the
apex of the cone up or down in theoretical space you could choose
whatever parallel suited the cartographer's need. The flattened shape
is awkward, but if you are making a series of smaller charts, they can
be made in the traditional square or rectangular
formats :
Further, Lambert discovered that you could project your cone
through the Earth's surface and "touch" at two reference parallels with very little distortion:
You just sort of "tamp down" the bulge of the earth between the
"standard parallels" and ignore the very small distortions, as in this
example (note the bottom of the legend):
Here's a common use of Lamberts. The table at the bottom of the legend
shows the series of charts, using a variety of "standard parallels" to
assure the minimal distortion. The lack of distortion in the Lambert
projection is especially important for aerial
navigation because it permits us to draw a straight line on the chart
which is truly represents a great circle on the spherical earth, and
great circles are the true path of the radio waves we use to establish
positions and courses:
Pedantically yours,
Bill Butler
Professor of Aviation Technology (retired)
PART II - Steve's Questions and Bill's Response
Steve: Can you explain a little more about the Lambert Charts? Like, what are they
for? Who uses them?
Bill: Actually, they are used for a lot of things, but aircraft
navigation may be most common They come in different scales and with
differing levels of detail, but their use is universal, and even with
electronic mapping in most airline cockpits, these
are the charts which are digitized. The curved parallels of latitude
are a natural outcome of projecting a sphere onto a flat surface, but
they are only obvious over a great distance.
Steve: Why does it
say north to the right and south to the left when the map looks like
those should be east and west? Could use a more current one?
Bill: What you are looking at is
the
cover of an aeronautical chart, where the legend appears
explaining what is inside. "north/south" tells you that this is a
two-sided document and if you unfold it from this edge...
Steve: Why does it say Seattle, when it has the whole northern part of the US on it?
Bill: It [the map below] says "San Francisco", for reasons which seem obvious...at least to me
[;-> It is one of a set covering the entire nation and the little
map shows you what all the others are named. Note that the Lambert
standard parallels for this chart are different than
the ones used for the Seattle chart.
Steve: OK, this is the cover page of your map. And the little map that is cross hatched - in this case San Francisco - is what is inside this cover. And in the previous map, I now see that Seattle is also cross-hatched. And thus the north and south designations on the sides would make sense when I opened the map. Right?
Bill: Exactly
Bill: Let me back up a bit to Mercator, who shows that any "great circle" can
be projected without distorting it. He did this mathematically, but
graphically, it looks like a sphere within a cylinder. Note that the
when the sphere was inflated to contact all points
of the cylinder, only the equator stays exactly where it was, i.e.
undistorted. Now the meridians, while bent, retain their spherical
nature, just like the equator. That is, if you draw a straight line
from say Kodiak to Hilo, it is truly the shortest distance
and undistorted because they are both on the 150th Meridian, but if you
draw a line from Anchorage to Oslo on the map below, even they are both
roughly on the 60th Parallel, the track is hugely distorted, because as
we both know, the shortest way is over the
north tip of Greenland:
This problem can be attacked by what is known as a Transverse or
Oblique Mercator, that is by rotating the sphere within the cylinder so
that any line we choose is a "false equator, that is, the line of zero
distortion. This is actually how Pan American attacked
the problem when they began flying long overwater routes in the
mid-30s; they would make special charts specific to each route and the
crew could plot out their course knowing that what was on the paper was
the shortest line between two points on a sphere,
i.e. a great circle:
This solution is unworkable in a system where tens of thousands of
aircraft fly hundreds of routes every day, but Lambert, provides the
solution because we can use his theoretical cone (cones, actually,
because we choose the one which touches the surface in
the zone we are interested in) to map the entire earth with parallels,
meridians and course lines drawn to look straight on the flat paper.
This is what the inside of that Seattle aeronautical chart looks like. I
have drawn a line from Portland to Boise
in this example:
Notice that the meridians and the parallels appear to be straight (well,
the parallel does show a bit of bend, but the error induced is
miniscule), although they are segments of a great circle. Lambert's
genius is that the cartographer can select any "standard
parallel" for zero distortion, and when he makes the next map to the
south or north, choose a different parallel (actually two are used for
each chart with acceptable error).
Steve: Let's see if I got this. With the original Mercator charts, you're saying, if you just focus on a small portion, even though there is distortion away from the equator, the distortion is the same throughout that small portion, so it's still usable. Is that right?
Bill: Almost. There is a difference between the drawn line and the great circle which truly represents the shortest distance, but an 18th century sailor who was going from Newcastle to Bergen, unless he understood the spherical geometry (unlikely) would be unaware of any error, except it would take him a bit longer than he thought it should. If he was sailing from Cardiff to Cadiz, there would be almost no error at all because the north-south meridians drawn as straight lines are still great circles. Generally speaking, these errors were inefficient, but harmless until we started navigating over great distances in short times and using radio to determine courses and positions.
Steve: And Lambert, by changing the "equator" for each map, clears the distortion that way?
Bill: Exactly.
And I'm sure Bill would be glad to answer further questions in the comments.
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