ạ/;°ḊÆS,ÆẠƲP€SÆAHÆSḤ,Ʋ×⁽ßx
A dyadic Link that accepts the longitudes on the left and the latitudes, in the same order, on the right and yields the two distances, [direct, surface].
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How?
Labelling the two longitudes as \$\lambda_i\$ and the two latitudes as \$\phi_i\$, the angle between the lines from each of the two points and the centre of the sphere, the central angle, is[1]:
$$\Delta\sigma = \arccos{(\cos{|\lambda_1-\lambda_2|}\cos{\phi_1}\cos{\phi_2}+\sin{\phi_1}\sin{\phi_2})}$$
The shortest distance between two points in Euclidean space is a straight line, so we have an isosceles triangle with the equal side lengths being the sphere's radius, \$r=6371\$ kilometres, so the direct path in space is (using \$\text{opposite}=\text{hypotenuse}\sin\alpha\$):
$$d = 2r \sin \frac{\Delta\sigma}{2}$$
The shortest distance between two points on a sphere is the minor arc of the great circle on which they lie. This is the arc projected by the angle \$\Delta\sigma\$ and thus has length:
$$s = r \Delta\sigma$$
The code takes \$(\lambda_1, \lambda_2)\$ on the left and \$(\phi_1, \phi_2)\$ on the right and produces \$(d, s)\$:
ạ/;°ḊÆS,ÆẠƲP€SÆAHÆSḤ,Ʋ×⁽ßx - Link: Longitudes; Latitudes
ạ/ - reduce {Longitudes} by absolute difference
; - concatenate {Latitudes}
° - convert from degrees to radians
Ʋ - last four links as a monad f(X=that):
Ḋ - dequeue -> Latitudes
ÆS - sine -> [sin(p1), sin(p2)]
ÆẠ - cosine {X} -> [cos(|l1-l2|), cos(p1), cos(p2)]
, - pair -> [[sin(p1), sin(p2)], [cos(|l1-l2|), cos(p1), cos(p2)]]
P€ - product of each
S - sum
ÆA - arccosine -> the central angle
Ʋ - last four links as a monad - f(CentralAngle):
H - halve
ÆS - sine
Ḥ - double
, - pair -> [2 × sin(CentralAngle / 2),
CentralAngle]
⁽ßx - 6371
× - multiply -> [6371 × 2 × sin(CentralAngle / 2),
6371 × CentralAngle]
= [direct, surface]
