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# Geology/Latitude

Question
QUESTION: How are GPS coordinates converted to miles? Lets assume we're traveling only in one compass direction. Will the degrees of Latitude change the miles equivalent? How does that type of thing work?

This question can be answered with a little background in maritime navigation. Before we go into any detail here, one must understand the definition of a Great Circle in navigating the earth.
So, what is a great circle? A great circle is the largest circle you can fit into a sphere. If you cut a sphere along a great circle, you would get two identically sized halves. Incidentally, it also describes the shortest distance between two points on a globe. On a flat plain, the shortest distance between two points is a straight line. On a globe, the shortest distance between two geographic locations lies along the perimeter of a great circle placed through these two geographic destinations. There are a few natural great circles on a globe. The equator, for example is a natural great circle. It cuts the earth into precisely equal halves of the Northern and Southern hemispheres.
Also, each longitudinal line completes a great circle if you add the back of the line (circle) to it, for example the Prime Meridian (0 degrees longitude) plus the back (the 180 degree longitude line) which combined would make a great circle. If you would cut the earth along the Prime Meridian and the 180 degree line, you would get precisely equal halves of the earth called the Eastern and Western hemisphere. Or try the 25 degree East longitude line, for example, which at its back has the 155 degree West longitude line and together they would make a great circle. ALL longitudinal lines with their back counterparts describe great circles and would cut the earth into exactly equal halves.
Along any great circle on the surface of the earth, distances can be recorded in a linear measurement such as miles or kilometers. However, since it is a circle, distances can also be recorded as angular measurement. If you locate to geographic points along the perimeter of a great circle, these are obviously certain miles apart. But you can also measure their distance by drawing a straight line from both of these locations to the center of the circle and then measuring the angle between these straight lines at the center of the circle.
Mariners (and aviators) use the distance measurement of nautical miles (1 nautical mile = 1.151 statute or land miles or ‎6,076 ft). The reason being is that nautical miles convert directly into angular distances along any great circle and visa versa. Each degree angular distance is exactly 60 nautical miles along any great circle, no matter how you place it. (Which also means that 1 nautical mile is exactly one minute of arc along any great circle). Therefore angles can be easily transposed into nautical miles. And then these nautical miles can be converted into land miles or kilometers. Let me illustrated along a natural great circle. If we take two geographic locations along the equator for example we can easily convert longitude and latitude position into a distance. Lets use N 0 degrees and W 25 degrees as one location and N 0 degrees and W 123 degrees as the other location. (N 0 degrees for both locations because we are at the equator). The difference between 123 degrees minus 25 degrees is 98 degrees. Now 98 degrees angular distance times 60 (remember 60 nautical miles per degree) gives as a linear distance of 5880 nautical miles or converted to land miles = 6767.88 mi.
This is a piece of cake and you can do this in your head if you stay along any natural great circle. If you have an oblique great circle slicing through the earth at an odd angle, lets say along a great circle connecting Sydney, Australia and San Francisco, California, things get a little more complex and some advanced trigonometry has to be used. There is something called the 'haversine' equation to calculate the great-circle distance between two points – that is, the shortest distance over the earth’s surface – giving an ‘as-the-crow-flies’ distance between the points. If you are into math, here is a wikipedia brief on the calculation: https://en.wikipedia.org/wiki/Haversine_formula
If you are not into math, it sufficeth to say that it is all about great circles. Once the great circle is established, the angular distance can be extracted. All one has to do is to convert the angular distance into linear distances.
Hope this helps!

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QUESTION: So if I had two coordinate positions not on a great circle I would need to somehow use that advanced math problem to determine exact land miles?

Hi James,
Actually you would generate a great circle that would have both of your coordinate positions on it. This great circle is created mathematically when using the Haversine equation. The equation then translates the accurate angular distance in degrees, arc minutes, arc seconds and the decimals of an arc second between your two coordinate positions along this generated great circle into an exact linear distance. Nautical miles would be a very straight forward translation with every arc minute being exactly 1 nautical mile and every arc second being exactly 1/60th of a nautical mile.
You can sort of simulate this approach with a string on a globe. If you place the string tightly between your two coordinate positions on the globe, the string touching the globe between these two positions actually describes a part of a great circle going through these two points. The funny thing about strings on globes, the tightly held string on the surface of a globe held between two positions will always try to generate a great circle. The length of the string between your two positions is analogous to the linear distance between these two points. And to make it more clear, as there is only one straight line between two points on a flat surface, there is one and only one great circle that will exactly fit between two coordinate positions on a globe.
The Haversine equation is a mathematical model that does all of this for you, placing the one and only great circle along your two coordinate points and then directly translating the angular separation of your two coordinate positions into an exact linear distance along that great circle.

#### Uwe Richard Kackstaetter, Ph.D. (Dr.K)

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