0

This question is more about math than programming. I am programming a function which takes a square of geographical distance between 2 points with known latitude and longitude as an argument. There's a straightforward way to compute it: calculate dot-product, then take arccos, and multiply by Earth radius. Then square the result and you get the square of geographical distance assuming Earth is a sphere (which is acceptable approximation in my case).

However I would like, if possible, to avoid an expensive arccos() call, especially given that I can easily obtain the square of the tunnel distance (by either Pythagorean theorem or the dot product).

I also read here http://en.wikipedia.org/wiki/Geographical_distance#Tunnel_distance about underestimation formula which I can use to get tunnel distance from geographical distance. In my case however, I need the opposite (tunnel to geographical), and for the square. I played with Taylor series and got a rough approximation:

G square = T2 / (1 - (T2/R2)/12.0) // here G2 is square of geographical distance, T2-square of tunnel, R2-square of Earth radius. I also was able to get a more accurate formula:

G square = T2 / (1 - (T2/R2)/12.0 - ((T2/R2)^2)/240.0).

This last formula gives error of only 3.8mm for G=1000 km, and less than 50cm for G=2000 km.

However, I still cannot mathematically prove this formula, at least when using Taylor series. Wonder if it's possible to get the mathematical proof and also expansion of this formula for larger values of G/T. Thanks!

antonio
  • 375
  • 4
  • 13

1 Answers1

0

Why tunnel distance from geo distance?. There is no geo distance. There are many possibilities to calculate a distance between two points on earth.

Just take the two lat/lon cooridnates, and then calculate the distance between them using a simmple cyclindrical projection.

This needs only a cos(centerLatitude), and a multiplication with a factor. (meters_per_degree)

See also Cyclindrical equi distant projection. Up to some kilomters (abou 10 to 100) this gives sufficient accuracy.

AlexWien
  • 28,470
  • 6
  • 53
  • 83
  • What is your accuracy for distances say 500-1000 km with suggested formula? How about polar regions? – antonio Jul 10 '14 at 18:56
  • most applications don't need polar regions, 500km - 1000km will be much to long for that cyl equidist projection. If you need polar region and or correct distances for high distances then you could use the e haversine formula, which is well suited, especially for short distances but is a bit slower than the greater circle formula. Both shoul dhave an acuracy within 1%, If you need more acurate use vicencies iteration. – AlexWien Jul 10 '14 at 19:27
  • Perhaps I should have explained better. I am interested in a fast approximation which preserves good accuracy (1% is not enough) for square of geo. distance given tunnel distance. Not restricted to any particular region. Why not use haversine or Vincenty's formulas? Because I am calling it 1000s if not millions of times and would like to speed it up. The question also has theoretical importance. The formula I have given gives excellent, fast result for distances up to 1000km in any region. I would like to know if it can be theoretically explained and expanded for even greater distances. – antonio Jul 10 '14 at 20:02