[EM] Maps on which to draw rectangular districts
MIKE OSSIPOFF
nkklrp at hotmail.com
Thu Jan 8 03:13:02 PST 2004
Lines of latitude & longitude are, as I was saying, the obvious &
natural borders for the rectangular districts. Those lines appear as
straight
lines on cylindrical projections. The Mercator, Miller, and Gall's are
cylindrical, but, for displaying the districts with latitude/longitude
grid borders, the equirectangular, or cyllindirical equidistant, would be
best.
Another cylindrical projection, the equal-area "Peters" projection would
show the district areas in their correct proportions, though that may not be
important. Actuallly the Peters projection wasn't invented by Peters. It
goes back to Lambert, in the 18th century. Peters merely named it after
himself. Regrettably, he recommends and sells it as a world map. As a
world map, it grotesquely distorts the shapes of the continents. As one
cartographer pointed out, on the Peters world map, Africa & South America
look like tattered long underwear hanging out to dry.
The Peters map distributors imply that Peters is the only equal-area
projection, that equal-area is a Peters projection innovation. But equal
area projections were in use at least back to the 15th or 16th century. In
fact they go back to Ptolomy, if you allow for the fact that Ptolomy used an
approximation for longitudes, very justifiable considering how poorly
places' longitudes were known in Ptolomy's day.
Because it's equal area (like lots of other maps) the Peters people emphsize
how their projection is the one that's fair to the 3rd world, pointing out
that the Mercator magnifies the extreme North.
But it's no fairer in that regard than all the other equal area maps. And
its ridiculous shape-distortion of tropical continents makes it really a
joke to say that Peters is the map that's fair to the tropics.
Regrettably, reminiscent of success of the IRV promoters, the Peters
projection people have managed to convince lots of people that theirs is the
map that's fair to the people of the 3rd world, a powerful incentive that
has caused many organizations to adopt that projection, and which results in
it being the only equal area projection that you can find in a store.
How to recognize the Peters projection: Africa & South America are absurdly
long and skinny.
Well, the topic of districting opens the topic of geography.
Another possibility would be to use a map that are officially used to
map the state, on which to draw rectangular districts. So the borders are
then straight lines on that official map, instead of being latitude &
longitude
lines. Official maps tend to be conformal, and, as a result, the right
angles
at the corners of the districts will result in right angles at the corners
of the districts on the ground. That will be true of conformal projections
and cylindrical projections, and of districts whose borders are latitude &
longitude lines.
Maybe it would be desirable for the district borders to be straight
lines, as reckoned along the ground--great circles, the shortest distance
between 2 points if the Earth is assumed spherical.
That can be achieved if the rectangular districts are drawn on a map
that uses the gnomonic projection.
Earlier I said that that map radically distorts distances and areas, by
which I meant that scales can vary greatly on such a map. But actually,
for an area the size of a state, that scale variation won't be a problem,
and so the gnomonic would be a good choice for the map on which to draw the
rectangular districts, if it's desired that the district borders be
straight lines along the ground. Such straight-line borders could be a
great convenience for surveying the districts.
Another possibility: Surveyors in each state use a state co-ordinate system,
based on rectangular co-ordinates on a particular map. If that same map is
the one on which the rectangular districts are drawn, then the district
boundaries will be lines along which one of the state co-ordinates is
constant. I suggest that possibility because it might be convenient for
surveying or official mapping.
These maps are conformal, and so the corners of the districts, on the
ground, will be right angles.
There are obvious ways to write a simple rectangular mapping formula. If the
state is L times as long north-south as east-west, and N is the number of
districts, then, divide the state into sqr(NL) bands of east-west extent.
Divide each band into rectangles containing population equal to the state's
population divided by N. The band will end with a fractional-size district.
Adjust the north-south dimension of the band to make an integer number of
whole districts of the desired equal population.
Obviously, adjust to a larger north-south dimension, to add a district to
the band, if the fractional district is more than half of the desired
population. Adjust to a smaller north-south dimension, to subtract a
district from the band, if the fractional district is less than half of the
desired equal population.
As I said, simplicity is the most important consideration for the
district-drawing formula. Rectangular districts, drawn on some specified
map, by a simple formula such as I've described, are the way to draw the
districts.
Mike Ossipoff
Mike Ossipoff
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