[EM] Districts from Gnomonic map have virtually right-angle corners

[MIKE OSSIPOFF]

Ernie--

You mentioned the advantage of a conformal projection, which correctly 
reproduces angles.

But though the gnomonic isn't conformal, districts that result from 
rectangles on a gnomoniic are practically indistinguishable from 
rectangular. Their corners differ from a right angle only by a tiny amount.

Even if the entire U.S. were mapped with the gnomonic projection, centered 
at the center of the country, a district, at one extreme end of the country, 
at the periphery of the map, a district which is square on the gnomonic map 
will, on the ground, have corners that only differ from right angles by 
about 1.4 degrees. That's if the district has one of its corners pointed 
toward the center of the map. If the district has one of its _sides_ pointed 
toward the center of the map, and if it  has the average area that national 
senate districts would have if the senate had districts, then it's corners 
will differ from a right angle by only a fraction of a degree.

And if,  instead of a national map, it's a state map, even for a state with 
great extent, like California, the districts' corners will only differ from 
a right angle by a fraction of a degree, regardless of how they're oriented 
with respect to the center of the map.

So, to have districts whose corners don't differ perceptibly from a right 
angle, it isn't necessary to use a conformal projection or a cylindirical 
projection.

Districts whose boundaries are latitude/longitude llines will have corners 
that are exactly right angles.

The longitude lines, meridians, are straight lines on the ground. The 
latitude lines, parallels, are circles on the ground--constant curvature. If 
you were driving a road along a parallel, you'd have the steering wheel at a 
constant position to drive along the parallel.

Longitude/latitude district lines, the geographical earth co-ordinate 
system, is an appealing way to make rectanglar districts.

The all-straight district lines made by rectangles on a gnomonic projection 
are appealing too

Previously I'd said that the gnomonic would give noticibly unsquare 
districts when a country as big as the U.S. is mapped. Not so. I'd initially 
believed that because I'd seen gnomonic maps showing nearly half of the 
world, and maps of such a large area can have great area exaggeration and 
shape distortion at the periphery, especially for large shapes. But that 
isn't true of the gnomonic when it's mapping a country. Well of course a 
country as large as Russia would have somewhat more departure from 
rectangularness than the U.S. would.

Mike Ossipoff

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