# Great-circle distance

The great-circle distance is the shortest distance between any two points on the surface of a sphere measured along a path on the surface of the sphere (as opposed to going through the sphere's interior). Because spherical geometry is rather different from ordinary Euclidean geometry, the equations for distance take on a different form. The distance between two points in Euclidean space is the length of a straight line from one point to the other. On the sphere, however, there are no straight lines. In non-Euclidean geometry, straight lines are replaced with geodesics. Geodesics on the sphere are the great circles (circles on the sphere whose centers are coincident with the center of the sphere).

Between any two points on a sphere which are not directly opposite each other, there is a unique great circle. The two points separate the great circle into two arcs. The length of the shorter arc is the great-circle distance between the points. Between two points which are directly opposite each other (called antipodal points) there infinitely many great circles, but all have the same length, equal to half the circumference of the circle, or [itex]\pi r[itex], where r is the radius of the sphere.

Because the Earth is approximately spherical, the equations for great-circle distance are important for finding the shortest distance between points of the surface of the Earth, and so has important applications in navigation.

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## The formula

Let δ1 and φ1 be respectively the latitude and the longitude of the first point, and δ2 and φ2 those of the second. Let r be the Earth's radius. Then the great-circle distance is:

[itex]d(\delta_1, \varphi_1, \delta_2, \varphi_2) = r \cos^{-1}\left(\sin\delta_1\sin\delta_2 + \cos\delta_1\cos\delta_2\cos(\varphi_1 - \varphi_2)\right).\,[itex]

This equation can have large rounding errors when the distances between nearby points are computed, however, so it should not generally be used in practice. Instead, an alternate formula, mathematically equivalent but with less rounding sensitivity, is:

[itex]d(\delta_1, \varphi_1, \delta_2, \varphi_2) = 2r \sin^{-1}\sqrt{\sin^2\left(\frac{\delta_1 - \delta_2}{2}\right) + \cos{\delta_1}\cos{\delta_2}\sin^2\left(\frac{\varphi_1 - \varphi_2}{2}\right)}.[itex]

For more explanation, and a slightly different version of this formula, see the haversine formula.

## Spherical distance on the Earth

The shape of the Earth more closely resembles a flattened spheroid with extreme values for the radius of curvature of 6336 km at the equator and 6399 km at the poles. Using a sphere with a radius of 6367 km results in an error of up to about 0.5%.

## A worked example

In order to use this formula for anything practical you will need two sets of coordinates. For example, the latitude and longitude of two airports:

• Nashville International Airport (BNA) in Nashville, TN, USA: N 36°7.2', W 86°40.2
• Los Angeles International Airport (LAX) in Los Angeles, CA, USA: N 33°56.4', W 118°24.0'

You will have to convert these coordinates to a more mathematically friendly form using a simple method before you can use them effectively in a formula. After conversion, the coordinates become:

• BNA: [itex]\delta_1[itex] = 36.12°, [itex]\varphi_1[itex] = −86.67°
• LAX: [itex]\delta_2[itex] = 33.94°, [itex]\varphi_2[itex] = −118.40°

You'll need to convert these coordinates to radians instead of degrees for them to be useful in the formula:

• BNA: [itex]\delta_1[itex] = 0.6304, [itex]\varphi_1[itex] = −1.5127
• LAX: [itex]\delta_2[itex] = 0.5924, [itex]\varphi_2[itex] = −2.0665

Now you can simply substitute numbers in the formula above:

[itex]d(\delta_1, \varphi_1, \delta_2, \varphi_2) = r \cos^{-1}\left(\sin\delta_1\sin\delta_2 + \cos\delta_1\cos\delta_2\cos(\varphi_1 - \varphi_2)\right)[itex]
[itex]{}\cong 0.45306\,r[itex]

Substituting r with 6367 kilometers, we get:

[itex]{}\cong 2884.6\;\mathrm{km}[itex]

The distance between LAX and BNA is then 2884.6 km or 1792 miles.

## Spherical coordinates

In the spherical coordinates used by mathematicians and physicists, usually when considering other spheres than the Earth's surface, the great-circle distance is found as follows. If φ is the azimuthal angle and θ the colatitude then the spherical distance is given by

[itex]d(\theta_1, \varphi_1, \theta_2, \varphi_2) = r \cos^{-1}\left(\cos\theta_1\cos\theta_2 + \sin\theta_1\sin\theta_2\cos(\varphi_1 - \varphi_2)\right).[itex]

Note that if φ1 = φ2 then the distance formula reduces to d = r1 − θ2|.

The notation above is that used in the physical sciences. Pure mathematicians, by contrast, conventionally interchange the roles of the letters φ and θ.

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