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Geography redux

2004-12-07 17:14:34.490543+00 by Dan Lyke 3 comments

Since that previous "you've got to be kidding" elicited at least one "tell me more", another update on geography:

Lattitude and longitude are angles from some "center" position somewhere down near the middle of the planet. There are three ways to get that "center" position:

  1. Take straight down, like with a pendulum, from your current position, go some known number of miles (ie: "the earth has radius X at sea level, we're at sea level + Y") in that direction, and call that center.
  2. Take the axis of rotation of the earth (say, gauged off the stars) for the center of longitude, and something judged off the motion of the sun over the seasons for the center of lattitude.
  3. Take an imaginary ellipsoid (ellipse in 3d), try to make that best fit the known surface of the earth, and then use the geometric center of that ellipsoid.

That first option is handy, you can calculate most of what you need from right where you're standing. Unfortunately, the inside of the earth isn't homogenous, all the same consistency or density, so if you take a bunch of plumb lines from different places on the earth, they won't converge at a single point. You'll accumulate error because a big hunk of granite near the surface in your vicinity perturbed "down" ever so slightly.

The second solves that center issue, because you're finding center based on the entire mass of the earth. Of course this might move around a bit as, say, someone on one side of the globe builds a reservoir up in the mountains.

The third is handy once you've discovered space flight, but it makes sea level, the reference point for so long, no longer a good measure of altitude.

The obvious problem with the third is inherent to the other two as well: The earth isn't spherical, so even if 90 degrees and 0 are always at the same place because you've chosen the same center, the intervening positions can be off.

But that's okay, because up until recently we've been mostly concerned with positions in one region. So we'll use local systems that maybe match up with another system in a place that's in the center of where we're interested (Kansas got a lot of attention), and then use an ellipsoid that matches our region of interest more closely, and ignore the fact that it's a little different than the other system around the coastlines or the edges of our region of interest.

What this means is that when you say "lattitude, longitude and altitude", this can be in one of literally thousands of the various permutations of those three ways of finding the center of the coordinate space combined with the various different reference ellipsoids combined with the notion of the "local zero", or where this coordinate space matches another.

The differences between these spaces can be hundreds of meters, especially in altitude. Luckily GPS[Wiki] works in one of those spaces, but only with an accuracy of about ten of meters in lattitude and longitude, less in altitude, and aerial photography, local maps, and surveying information may be in any one of the other spaces.

And then, because the local earth surface appears pretty flat to us, we want to map these things down to something in two dimensions on a nice handy grid where we can use a flat ruler to tell the distance between two points, and it gets really hairy.

[ related topics: Photography Aviation Space & Astronomy Graphics Maps and Mapping ]

comments in ascending chronological order (reverse):

#Comment Re: made: 2004-12-07 22:02:43.054202+00 by: petronius

It seems to me that the answer is to find some average of those reference points and just arbitrarily mark that as the zero point, something like the Meridian of Greenwich.

#Comment Re: made: 2004-12-07 22:35:01.554877+00 by: Dan Lyke

The point of the reference points is that they're a way to map between two systems. So, yeah, for the 1927 and 1983 standards in the U.S. (NAD27 and NAD83), Meade Ranch, Kansas is the common point, and that point is the same for both of them, but other places in the United States can be a tenth of a mile apart from one to the other.

#Comment Re: made: 2004-12-13 16:44:47.224567+00 by: ebradway [edit history]

For cadastral mapping (land ownership platts, etc), each state uses it's own "state plane" projection. State plane works well for horizontally long states, like Tennessee, but lose accuracy in vertically long states like California (which uses two different state plane systems). In each system, a control point is established and all coordinates are based on the distance from that point.

The best model for the shape of the Earth is a geoid - which is calculated by measuring the strength of the gravitational field across the surface of the Earth. Here's a great site at NOAA The problem with the geoid is that you no longer have a single center and your map projection transformations become unweildy.

And <href="http://www.colorado.edu/geography/gcraft/notes/mapproj/mapproj_f.html">here is one of the best sites for learning about map projections.