Wednesday, June 9, 2010

GPS Maps 101: The Numbers

[This post is part one of a multiple part series about understanding and using GPS coordinates in offroad travel: Latitude, Longitude and coordinate display conventions.]

One of the most universally known, but nearly always the least understood, object is — the circle.

All at the same time it is so elemental that the youngest of children can recognize it, turn around it and tell you that wheels, balls and clocks are all circles; but, so sophisticated that it nearly takes a PhD in mathematics to understand and use it.

It takes a bit more sophistication, but not much, to know that you can divide it into segments, either equal or not; and, that when the segments are uniform they "look right," like spokes of a wheel and numbers on a clock, when they aren't, they don't.

Ask any middle-school child to do a "one-eighty" and see if they understand or a skate-boarder if they can do a "three-sixty." There will not be a moments hesitation.

Very little after first grade do we learn that the big hand is half-way between one and two when it's "one thirty." But talk about "degrees" and you had better be referring to wearing a coat or eyes will roll and corneas will glaze over.

Why are there 360 degrees in a circle? …It's better if you just accept it and get on with your life; but, if you MUST know more, then it has to do with the mathematician Claudius Ptolemy (100-170 AD) and the Babylonians (who used a base 60 numbering system instead of base 10 like we do).  By trial and error they found that when you divided a circle into 360 equal units their math equations came out better. Like I said: it just is – so get on with it.

[Lest you think too poorly of the Babylonians - just remember that the number 360 is evenly divisible by 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360; and, that there are precious few other numbers as small which are so well endowed with "factors." Then there is also that time-relativity/compatibility thing – you know, 24 hours = 1 earth rotation. Thank you Greek's and Egyptians for 12 and 60/60.

Not to be out done, the British divided the circle into 400 degrees once – well, that's the British for you. They were also the ones who forever rub our face in their monarch's shoe size, the FOOT.]

Simple Conventions

Sometimes we can employ some simple conventions to make life easier. Not since high-school have we heard anyone claim "it's 180 degrees after 30 degrees." [A twelfth of 360 being 30°] It's much easier to say: "it's half past one.

You can immediately see that on a foot square clock, 12 divisions or 30 degree segments are pretty much what is needed – and still be readable. Extrapolating to the size of the earth, however, needs a tad bit more precision.

We'll talk more about the Earth not being round later, but for now lets assume that it is close. Being a circle, it can be divided into 360 DEGREE's. And each of those segments can be further sub-divided, like the clock, into SIXTY segments or MINUTES. And each of those sub-segments can be further divided into SIXTY sub-sub-segments or SECONDS. And each of those… well that's when we usually switch back to true decimals.

That's it. The actual point locations of any spot on a sphere can be "addressed" using degrees, minutes and seconds (all angular measurements) having NOTHING to do with distances! That's assuming you can agree on a starting point and a direction for drawing lines. Of course the origin point would be the true center of the earth.

Once people figured that the Earth spun like a top, it almost mandated one of the directions for drawing lines – around through the poles. And once that was out of the way, there truly was only one other direction in which the other coordinate line could be drawn – tangential to the poles, around the center.

So, we now have two dividable circles. One through the poles and one around the "belt." Tick marks could be drawn all along those circles which could provide pointers to every location on the earth.

If you run a piece of thread around the earth at the "belt" and tie it off. Then, another on top of that. And another, and another, until you get to the pole. You will then have the "lines of latitude," or "circles of latitude."  They are lines that are DRAWN East-West BUT when you describe which one of the lines you are on it TELLS YOU how far North-South you are on the earth. Therefore: LATITUDE = North-South location, and is sometimes labeled on maps with the greek letter Phi (φ - prounounced "fi" not "pi").

To finish out this analogy, use the rest of your spool of thread to run around and around the earth up and down in the north-south direction. Only this time rotate about the axis as you tie them next to each other and you will have drawn the "lines of longitude." They are lines which are DRAWN North-South BUT when you describe which one of the lines you are on, it TELLS YOU how far East-West you are on the earth. Therefore: LONGITUDE = East-West location, and is sometimes labeled on maps with the Greek letter Lambda (λ - pronounced "lamb dah").

Wile you have been doing all the string winding you probably have noticed a bunch of peculiarities. For example the circles of latitude getting smaller, and those of the longitudes bunching up, when either of them get near the poles. That, we'll try and make sense out of later; but, for now we'll just add it to when we talk about the earth not being round.

You may also notice that when we try to give a name to all those lines, as far as the North-South (latitude) goes we don't actually need to use the whole 360°, the line only goes 180° from the north pole to the south pole. In fact we only even need to use half of them, if we just say that "everything above the Equator is N or "plus" and everything below is S or "minus." That way we only need to go from N (or +) 90 to zero and back to S (or -) 90. Bob's your uncle.

Parallels and meridians

[W]hen people thought the Earth was flat, they were wrong. When people thought the Earth was spherical they were wrong. But if you think that thinking the Earth is spherical is just as wrong as thinking the Earth is flat, then your view is wronger than both of them put together.” Isaac Asimov
Vertical lines around the earth from pole to pole are called "Meridians" and there is an infinite number of them depending upon how fine your pencil is and how narrow you can draw lines. The horizontal lines all the way up or down, parallel to the "belt," are called "Parallels."

Of course, with the Earth, the Parallels are more circular than the Meridians, but more on that later. And one can draw an infinite number of Parallels all up and down the globe depending upon how thin you make the lines. The circles get smaller and smaller the further you go away from the "belt" until they… well… become, for all intents and purposes, a dot!

The numbering system has to start somewhere and the place which started and funded the "standardizing" of all this stuff got to name it after themselves – Greenwich England. The meridian running right through their town is "zero" or "prime" – the Greenwich or Prime Meridian [0° 0' 0"]. [Actually the half on the side of England is the Prime and the half on the other side of the globe is called the "International Date Line" [180°]. All those points to the west of Greenwich are labeled "W" or "-", those east are labeled "E" or "+".

The central "belt" line also has a special name – the Equator. And, you remember from junior high geography that two other lines, the Tropic of Cancer [23º 26′ 17″ north of the Equator] and the Tropic of Capricorn [-23º 26′ 17″ south of the equator] are the parallels marking the extremes of the sun's path across the sky with the change of the seasons due to its tilted axis.

What you may not remember is that there are two other major circles of latitude: the Arctic Circle [66º 33′ 43″ (or 66.5619°) north of the Equator] and the Antarctic Circle [-66º 33′ 43″ (or 66.5619°) south of the Equator.] Every place north and south of these lines respectively get to have a full 24 hours of day or night twice a year.

What sometimes confuses many people who look at a globe is that the tick-mark labels for the Parallels (longitude) are often placed on the Prime Meridian (a latitude line). Conversely the tick-mark labels for the Meridians (latitudes) are often found on the Equator (a longitude line).

Just remember, when you already know the Meridian (long) you are on, your position along that meridian is given by your Latitude. And, if you already know the parallel (lat) you are on, your position along that parallel is given by the Longitude.

Coordinate Writing Conventions

There should be only one other item we need to discuss to complete the explanation of what the numbers mean: the several different styles for writing the same coordinates.

Here are all the different styles for writing the coordinates of Falling Man Petroglyphs on the Gold Butte in Nevada.

Degrees, Minutes and Seconds: N 36° 30' 28"; W 114° 11' 14.7588"   — or —   36° 30' 28" N; 114° 11' 14.7588" W
  —   (using N/S/E/W prefix's. You could use + and -'s instead. Note also that the seconds are given as a real number, read: "fourteen and 7588 ten-thousandths seconds.")

Degrees and decimal minutes: N 36° 30.471′, W 114° 11.246′
    (using N/S/E/W prefix's. Note that the minutes are given as a real number, read: "eleven and 246 thousandths minutes.")

Or, since the advent of the computer with it's increased computational accuracy, and desire to avoid unnecessary conversions, the straight Decimal Degrees are used – especially by programs like Google Earth.

Decimal Degrees: 36.507857; -114.187433
   (Negative numbers for S & W with latitude given first separated by either a semi-colon or comma.)

[Yes, I understand that I've left out the UTM system. Rationale: first, it's an antiquated system but with just enough utility and prevalence to prevent it from dying; and, second its explanation is so foreign to this post that it needs its own discussion.]

That's it. Oh, there is a nifty conversion program found at where you can play around to your hearts content with conversions of all kinds.

Next time: The Earth is NOT round - Coordinate Systems.


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