Distances along great circles

Distances along Great Circles

Take two places X and Y on the same line of longitude, i.e. one place is due north of the other. Suppose X is due north of Y. When travelling north from Y to X, you travel along part of a circle of longitude that is you travel along an arc of the circle.

The diagram below shows two points A and B on the same circle of longitude.

In the figure above, the sector containing the arc AB subtending ∅, is shown.

Recall the formula for the length of arc.

If an arc subtends an angle θ at the centre of a circle of radius R, then the length of the arc is given by:

Arc length = (2π × R × θ) / 360

In the figure above, R is the radius of the Earth and θ is the difference between the latitudes of points X and Y.

Taking the radius of the Earth R to be 6,400 km, the formula for calculating the distance between the two places along the same meridian becomes:

Distance = (2π × 6,400 × θ) / 360

Which simplifies to:

Distance = 111.76 × θ (in km)

NB: Remember, to find the difference in latitudes, take account of whether the places are north or south of the equator. If they are all found in south or north, then subtract the latitudes. If one is south and the other north then add the latitudes.

Nautical miles

Distances are also measured in nautical miles. One nautical mile (nm) corresponds to one minute of latitude.

Let the difference in latitudes between two places be θ. The number of minutes in θ degrees is 60θ, since 1 degree has 60 minutes.

Hence, the distance between the two places, along the arc of longitude,:

60θ nm = (2π × R × θ) / 360 kilometers

If we divide both sides by 60θ and take R = 6,400 km, then:

1 nm = (2π × 6,400) / (360 × 60) = 1.852 km

Therefore,

1 knot is the speed of 1 nautical mile per hour.

Navigation Related Problems

Example 1

Find the distance between Alexandria (31°N, 30°E) and Kigali (2°S, 30°E).

Solution

Note that both places are on the same longitude. The difference in latitude is 33°.

Use the formula:

Distance = (2 × π × 6400 × 33) / 360 = 3,690

Therefore the distance is 3,690 km.

The difference in latitude is 33°, so the difference in minutes is:

33 × 60 = 1,980′

This is the distance in nautical miles.

The distance is 1,980 nm.

Note:

1,980 × 1.852 = 3,690, to 3 significant figures.

Hence the two answers are the same.

Example 2

A plane starts at (20°S, 30°E), and flies north for 4000 km. Find its new latitude and longitude.

Solution

The plane flies north, hence its longitude is unchanged. The plane starts south of the equator, and flying north. It may cross the equator, and so end up north of the equator. In this case the latitude south of the equator will be negative.

Suppose the plane has flown along x° of latitude. Then using the formula for arc length:

4000 = (2πR × x) / 360

x = (4000 × 360) / (2πR) = 35.8

So subtracting 35.8° from 20°S implies:

20° − 35.8° = −15.8°

A negative latitude south is equivalent to a latitude north. Hence the new latitude is 15.8°N.

Therefore, rounding the answer to the nearest degree, the new position is (16°N, 30°E).

Example 3

A plane flies north from (10°S, 30°E) to (27°N, 30°E) taking a time of 3 hours.

Find its speed, giving your answer in both knots and kilometers per hour.

Solution

The plane is flying along a line of longitude. Its change in latitude is 10° + 27° = 37°.

The number of minutes is 60 × 37 = 2,220′, so it has flown 2,220 nautical miles.

To find the speed in knots, divide 2,220 nautical miles by 3, giving the speed as 740 knots.

Recall that 1 nautical mile is 1.852 km, and 1 knot (1 nm per hour) is equal to 1.852 km/hr.

Multiplying 740 × 1.852 gives the speed as 1,378 km/hr.