Geo calculation notes.txt

Alex Reiner (geocoder):

The angle/distance stuff is complicated until the moment when it all suddenly makes sense. It's probably much simpler than you're thinking it is. You should probably take a look at the methods in my Calculations module: https://github.com/alexreisner/geocoder/blob/master/lib/geocoder/calculations.rb 

The first method calculates the distance of one degree of latitude (which is simpler than longitude). When you're translating radians (or degrees) to distance, think about it as moving along a Great Circle. This simplifies your mental picture to two dimensions, so you can use a simple circle circumference equations: 

C = 2(pi)r 

We know the radius of the Earth is around 6359 km so if we substitute that we get: 

C = 2(pi)(6359) = 39955 

which is pretty close to the Earth's circumference: 40075 km. 

To calculate the distance spanned by one degree of latitude you just divide by the number of degrees in the whole circumference, which is 

360: 39955 / 360 = 111 

So to calculate the distance spanned by x degrees of latitude you just multiply by 111: 

distance = 111x 

For longitude it's slightly more complicated because the degrees range from 111 km at the equator to almost 0 km at the poles. 
But really you just do the same thing you did for longitude and then multiply by COS(latitude). 

So, for example, the distance of one degree of longitude along the 40 degree latitude line: 

111 * COS(40) = 85 km 

When you're not moving precisely along a line of latitude or longitude it's obviously more complicated, although really you can split the "diagonal" distance up into two vectors and calculate the length of the "hypotenuse." 

Conceptually the key is to think of moving along a Great Circle. A certain number of degrees (or radians) around that circle will always be the same distance, but degrees of longitude are a different thing and you have to translate them into circle degrees before calculating distance.