Calculate distance given Latitude and longitude

By Abeeda Mohammed (M728) on Tuesday, November 28, 2000 - 05:00 pm :

Hi,
I would like to know the formula for calculating distance using latitude and longitude.

Thank you,

Abeeda

By The Editor :

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By Kerwin Hui (Kwkh2) on Wednesday, November 29, 2000 - 01:08 pm :

Do you know about spherical polar coordinates?

By Kerwin Hui (Kwkh2) on Thursday, November 30, 2000 - 12:05 pm :

OK, I can't find how to draw a diagram of spherical polar coordinates..., anyway, latitude is slightly different from the way it is done in spherical polar coordinates, so we'll need to substitute

θ +

latitude

= π / 2

(Sorry for using radians here... it is natural to use radians)

Anyway, in spherical polar coordinates, we have

${ds}^{2} = {dr}^{2} + r^{2} d θ^{2} + r^{2} \sin^{2} θ d φ^{2}$

and the geodesic on the sphere is:

$\cot φ = k \sin (θ - α)$

where $k$ and $α$ are constants, determined by the boundary condition. (BTW, this can be derived using calculus of variation)

Substitute and integrate for $s$ will give you the answer.

Kerwin

By Brad Rodgers (P1930) on Friday, December 1, 2000 - 12:47 am :

Alternatively, if you want to find it without using calculus, you can apply the cosine rule to the following triangle:

$a = ((\tan θ_{1} - \tan θ_{2})^{2} + 4 (\tan (φ_{1} - φ_{2})^{2})^{1 / 2}$

$b = (1 + (\tan θ_{1})^{2})^{1 / 2}$

$c = (1 + (\tan θ_{2})^{2})^{1 / 2}$

(using Kerwin's concept for $θ$ [latitude starts with 0 at the top, right?]) (you can also figure $b$ and $c$ with cosines, but oh well, it's already typed)

To find angle $A$ (opposite side $a$ ). This is the angle between the two points. So just divide this by 360 and multiply by $2 π r$ (or just multiply by $r$ if you're using radians). If you're proficient in calculus, you can probably use Kerwin's method more proficiently, but otherwise, this method should work, albeit with just a little bit of work...

Brad