Calculator: Great Circle Distance

Computing great circle distance between points P1 and P2 on a sphere width Radius R:

Used Formulas

The points are given in polar coordinates latitude $\varphi$ and longitude $\lambda$:

(1)
(2)
where'
 $\varphi_{i,deg}$ ' =' 'Latitude in degrees Nord. Negativ values for South. $\lambda_{i,deg}$ ' =' 'Longitude in degrees East. Negativ values for West.

Converting into cartesian vector format using the center of the sphere as (0,0,0):

(3)
(4)
width
where'
 $\hat v_i$ ' =' 'Unit vector from center of the sphere to point $P_i$ $| \hat v_i |$ ' =' 'Lengt of vector $\hat v_i$ $R$ ' =' 'Radius of the sphere

The unit vectors to the 2 points are:

(5)
(6)
where'
 $\hat v_i$ ' =' 'Unit vector from center of the sphere to point $P_i$ $\varphi_i$ ' =' '$\mathrm{rad}(\varphi_{i,deg})$ = Latitude in radian. Negativ values for South. $\lambda_i$ ' =' '$\mathrm{rad}(\lambda_{i,deg})$ = Longitude in radian East. Negativ values for West.

The cosine of the angle $\alpha$ between the vectors is:

 (7)

And the great circle distance $L$ between the Points is:

(8)
where'
 $L$ ' =' 'Great circle distance between $P_1$ and $P_2$ $R$ ' =' 'Radius of the sphere (earth) $v_{1,x}$ ' =' 'X component of the unit vector $\hat v_1$ from center to $P_1$ $v_{2,x}$ ' =' 'X component of the unit vector $\hat v_2$ from center to $P_2$
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