Tag: great circle

The path between two points A and B on a great circle is an arc of the circle.

The arc of a circle is given by l= (θ/360) x 2πr where r is radius of the circle from where the arc is cut from and θ is angle between the two radii that encloses the circle.

Considering a great circle to be circular around the earth, arc of any length made by point A and B on that circle will be given by (θ/360) x 2πR where θ is the angle subtended by the arc AB at the Centre of the earth and R is radius of the earth.

Great circles are all the longitudes and the equator.

The distance between two points along any great circle is either measured in Kilometers(km) or in nautical miles(nm). A nautical mile is the length of an arc subtending and angle of 1/60 of a degree (1/60)^o or 1 minute (1′) at the center of a great circle. see the figure below;

in other words; 1^o = 60 nm along a great circle.

Consider the figure below:

The arc length of AB=60 x θ nm.

Example

Find the distance between the following points in (i) km (ii)nm

(a) A(65^oN, 15^oE) and B(20^oN, 15^oE)

(b) A(65^oN, 15^oE) and B(50^oS, 15^oE)

(c) A(0^oN, 58^oE) and B(0^oN, 12^oE)

Solution

we represent solution of (a) and (b) by sketching the points as in diagram below

(a) The angle subscribed by the arc AB along the longitude 15^oE is given by θ = 65^o-20^o=45^o.

(i) length AB = (θ/360) x 2πR km = (45/360)x 2 x 22/7 x 6370 = 5003.635km

(ii) arc length in nm = 60 x θ nm hence length AB= 60 x 45 nm = 2700 nm

(b) (i) The arc AC subtends angle 115^o = 65^o+50^o at the center of the earth. The arc length is therefore (115/350) x 2 x 22/7 x 6370 = 12.787.06 km

b(ii) length of AC in nm = 60 x 115 nm = 6900 nm

(c) point D and E are on the equator which is a great circle as shown.

The arc DE subtends angle θ = 58^o -12^o = 36^o at the center of the earth.

(i) Length of DE = (θ/360) x 2πR = (36/360) x 2 x 22/7 x 6370 = 4 002.91 km

(ii) DE in nm = 60 x 36 nm = 2760 nm

Example 2

An plane flew south from A(60^oN, 45^oE) to a point B. The distance covered by the plane was 8000 km. Determine the position of B taking π as 22/7.

solution

The relative position of position A and B cab be represented in the following sketch.

Let the latitude difference between A and B = θ

the length AB = (θ/360) x 2 x 22/7 x 6370 =8000

θ = (800 x 2520)/(44 x 6370 = 71.93^o

Latitude of B = 71.93^o – 60^o

hence the position of B = B(11.93^oS, 45^oE)

Related Topics

• Introduction to longitudes and latitudes
• Introducing latitudes
• Introduction to Longitudes
• Position on Earth’s surface

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