Monday, July 19, 2010

GPS Maps 105: Northing, Easting and Mercators Oh My

[This post is part five in a multi-part series about understanding the GPS system and its coordinate grid.]

Believe it or not both of these maps at the top of the post are REAL projections of the world map. I would hate to have to use them to ride offroad on my ATV but real projections none the less – just not the kind we're going to talk about today. In dutiful compliance with readers who have so far chided me for "ignoring" the UTM Coordinate system, this post is all about it.

However, I must confess that I sort of feel like a "lamb going to the slaughter" because I don't promise that after we are through you will understand it any better than you do now. I know that I don't think I do, and I've been studying the stuff almost non-stop for five months.

If you thought the last post about spheroids and datums was tough – hang on, this may be a bumpy ride! You almost need to forget everything we've talked about so far (except that the earth is not round) because it's all different as we will soon see. And the bass-ackward differences couldn't be anything except deliberate.

Your best bet during this discussion would be to keep reminding yourself that this UTM stuff was invented back when most of our worlds were what we could display on one map – and did not need to be concerned about the 'big picture.'

Just keep your mind focused that you are trying to merely solve the problem of looking at YOUR journey in YOUR corner of the world, draw it on a paper making it "look like" the ground that YOU were on, be able to easily visualize and calculate YOUR distances and YOU don't even know that there are "THEMS"… much less that there is an Australia.

Projections and Nomenclature:

There are several ways to refer to a coordinate system. Some people flippantly call ANY coordinate system a "projection," but this is not strictly true. "Geographic" coordinates, latitude and longitude, are  not considered "projected" at all.

They are calculated mathematically from triangulation measurements using equations which are as close to "true" as the current state of science will allow and cover the entire earth. This is the kind the "electronic" and GPS world is making ubiquitous.

On the other hand, most true "projected" systems used on the earth consist of two coordinates: nearly always called an "easting" and a "northing." [i.e. East of "something" and North of "something" similar.]

These are "measured" – so-to-speak – using tape measures, surveyors equipment and trigonometry; then, laid out on FLAT pieces of paper standing in for a single portion of the curved surface of the earth which is so small that it, hopefully, makes the scaling errors insignificant. This is the kind which sailors and explorers used before the space age when they didn't have anything else.

A CRS – Coordinate Reference System [or, SRS – Spatial Reference System] encompasses both geographic and projected. If a CRS is geographic, it may be called a GCS; if projected, a PCS.

Therefore, the two common types of CRS are: Latitude-longitude ("Geographic"), and Universal Transverse Mercator (UTM).

On a "foldable" map the spheroid lines of latitude and longitude need to be "projected" onto flat surfaces. This always causes distortion and the only question is how much and of what kind.

Until rocket capsules, absolutely nothing one tried to do to display a map without distortion worked every time. So there were literally hundreds of different map projection types invented – each attempting to minimize a particular type of distortion for a particular usage. Each having its advantages and disadvantages.


The most common projection, the Mercator, is the map for mariners who desperately needed to be able to know the distance they traveled and be able to draw a line on the map for their journey.

"Mr. Mercator's" invention projected the earth's geography onto a cylinder wrapped around the earth's equator so it preserved length. Therefore all Latitude and Longitude lines intersect at 90 degrees and do NOT get closer together at the poles as they should.

The Mercator is called a conformal projection because angles and small shapes on the map look like they do on the actual earth. HOWEVER, and it's a BIG however, this so-called "conformation" is only at the price of huge errors in scale at the edges of the map.

Did you ever pay much attention in school to how Greenland looked on the Mercator map at the front of the class? It seems huge – and looks as big as all of South America. [Truly only 1/8th the size.]

If you wrap a paper cylinder around the equator then project lines of latitude onto it using a mathematical formula (projection), when you unroll it, and lay it flat you've got a map with very little distortion traveling east and west. The distortion increases as you move north and south only. Areas close to the equator exhibit little distortion either way.

Transverse Mercator:

The problem with this approach is that most of the equator (where there's almost NO distortion) is over the ocean, and nearly all the area that most of us care about mapping is… not! Which leaves us with intolerable distortion over everywhere we care about.

So, even though it's a bit goofy, one solution is to turn the cylinder on its side and wrap it around a "meridian" – vertically instead of horizontally – in what is known as a Transverse Projection.

If you stop there, that isn't any solution at all – for anywhere except immediately adjacent to the meridian you choose. So you need to draw multiple maps. One for every "central" location you are interested in.

That way you have a whole set of maps, all centralized on meridians, rotated slightly along the equator. You just need to pick the one which is lined up with your ATV route.

Many countries and US states use Transverse Mercator for their official grid systems, especially countries such as New Zealand, which are long N/S and narrow E/W.

Universal Transverse Mercator (UTM):

Now this is all well and good while you offroad around in your own little world, using your own maps, making your own measurements. But what about when your buddy in another state wants to know about a ride you took.

You would pretty much need to send him your whole briefcase so he could use YOUR maps and equipment. HE would have never seen any of your stuff because he uses HIS OWN map.

Enter the Universal Transverse Mercator. What if you drew 60 meridians around the globe (every 6 degrees) and created a map with the line in the center? At least then everyone would be using the same kind of map!

But would you then be able to lay his map next to yours on a table and follow the trail? NO! Because they are a completely different datum and each one has so much distortion on the edges that they can never match up!

If you absolutely needed to combine the maps, the only way to do it is to draw a completely separate third map with it's center between the two and "re-project" both of your maps onto the third – which would then have it's own distortion at ITs edges.

But at least it was a start and that's exactly what they did. Then to minimize distortion in the North-South direction they further divided the globe into 8 degree slices going North-South from the equator and drew separate maps for those.

You must realize, of course, that this system is so dysfunctional at the poles that people just chopped off everything above 84° North latitude and below 80° South Latitude – let the penguins and polar bears get their own maps.

Labels and Measurements:

Now how do we label these 1200 maps that we've generated? Well, the creators were focused at the time on an interesting coincidence of measurement standards.

The Meter – was (until 1983) defined as: 1 ten millionth of the distance between the equator of the earth and its north pole! So the meter was the "built in" standard of measurement for every map, and seven digits is all that is needed for the north-south measurements.

That does mean that the northern and southern hemispheres need to be considered separately; but, hey remember that we don't even know THEM down/up there.

The zones are labeled from 1 to 60 beginning at 180° running EASTWARD [as opposed to longitude which is numbered westward]. They also carry a letter for their 8° segment north and south – this time beginning at the bottom in the southern hemisphere at "C" and running northward to "X" leaving out I and O [which look too much like 1 and 0]; but, leaving in "N" and "S" [which too often are confused with (N)orth and (S)outh].

The US is covered by map zones 10-19 and R-U. However, not able to leave well enough alone, there are 69 zones which have been altered slightly from the "standard." Sixty of them are around the "X" zone, which have all been extended 4° northward up to the 84th parallel. The other nine exceptions are all around Norway, three discarded and the rest expanded.

So, what is the benefit of all this? Well, with all these north and south measurements now in standard meters, calculating distances between two points (as long as they are on the same map) is a snap of the fingers to do in the field using the Pythagorean theorem compared with all that sixty-based, trigonometry latitude and longitude stuff! And, besides, who else is there to care? This is OUR map, there is no THEM.


All laid out flat on the table in meter measurements these maps just beg for you to whip out your ruler and measure them – in metric, of course. Beginning at zero on the equator and running to 10 million at the poles!

And that's what you would get on your SUVs odometer if you could run it up a meridian to the northern edge of the map – 9,328 kilometers at the 84th parallel. Piece-o'-cake, eh? The southern hemisphere… not so much!

Take a look at your UTM map. The southern hemisphere is cattywampus, beginning at 10 million on the equator and running to zero at the south pole. What's up with that?

I know that we said the Equator was zero and was numbered outward toward the poles to 10 million – minus 10 million in the south. But remember these are the original offroaders we're talking about who tried not to pack suitcases with Univac's, if they could help it.

Doing math watching out for negatives is about as much fun as reading your map upside-down so zero can be at the bottom. I mean the number is called a "Northing" not a "Southing."

As clever as they were, they decided to simply add 10 million to every Northing number in the southern hemisphere (they can do that because these are all separate maps remember) which effectively makes everything a positive number again. They call it a "false northing" which serves the purpose of making good old "Zero" always at the bottom of the map.


The map makers pulled the same kind of "magic" with the easting's as well.

Remember, each zone is 6° apart with the meridian running down the CENTER. That means each zone has both negative and positive easting's and offroading east or west across zones makes you flip from negative numbers going down to positive numbers going up every three degrees!

So, the convention is to give a "false easting" of 500,000 meters to each map (zone) making the numbers always positive. Any location west of the central meridian will have a number less than 500,000 and anything east (in that zone) more than 500,000.

Unfortunately, as the parallels get smaller away from the equator the actual numbers vary from map to map. Along the equator → the easting's range is from 167,000 to 833,000 meters on any one map. Obviously the ranges narrow the farther you get toward the poles; but, hey, the center is always 500,000.

Conventions In The UTM System:

We've described the layout of the UTM system. There remains to cover the conventions for writing coordinates so we can correlate the numbers to what we actually see on the maps.

To describe any one location on the globe you must begin with knowing which zone and which hemisphere you are in, then the easting and northing within that map. The zone number we have already covered and it is usually given first in the coordinates.

The hemisphere has a couple of different methods of display. Plus for north and minus for south, "N" for northern and "S" for southern (very poor practice which causes endless confusion), or the 8° zone letter we have also covered (the best and most usual method.)

For Khota Circus Petrolgyphs on Gold Butte in Nevada, the UTM coordinates are: 11S 0750768 4040522. The zone is 11 and in this case the designator is "S" not for "south" but for grid designator "S" (see how confusingly sloppy using the N(orth)-S(outh) terminology is?)

The remaining numbers are: 1) in meters; 2) given easting first then northing; 3) usually written with an even number of numbers (even if needing a leading zero); and 4) the last seven digits are the northing (which needs them), the rest are the easting (which doesn't always).

Because the figures are all meters, a decimal place can be added before the last three digits of each number to convert it to kilometers: 750.768 and 4,040.522 respectively.

That's it. All that remains is to get out your ol' map and look for the UTM legends.

Me… I think I'll just stick to my GPS and Google Earth… I know people in Australia.


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