Great Circle Formula: Concept, Applications, & Solved Examples

Great Circle is the largest possible circle that can be created by a plane that intersects a sphere. This plane forming the great circle always bisects the sphere through the centre irrespective of its angle of inclination. However, if the plane intersects the sphere without passing through the centre then it forms a small circle. The great circle is also called the Riemannian Circle or the Orthodrome. 

Key Terms: Great circle, centre, bisect, distance, sphere, arc, earth, radius, plane, point

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Concept of Great Circle

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Ideally, the shortest distance between two points on a plane surface is a straight line but the same is not true for two points on the surface of a sphere. The shortest distance between two points on the surface of a sphere is represented by a minor arc along the great circle passing through these two points and it is called the great circle distance or the orthodromic distance. 

Orthodromic distance

Equator as Great Circle and small circles parallel to it as latitudes

If we consider Earth as a sphere, then the great circle along the surface of earth passing through its centre will have a radius equal to the radius of the earth. The equator will be the great circle bisecting earth into two equal halves and the meridians will be the great circles passing through the north and south poles. 

Longitudes

For navigation purposes, the course followed by a ship is along the rhumb line which intersects all the meridians at the same angle. Although the rhumb line is not the shortest distance between two points, it is used for ease in navigation as it gives a uniform compass heading.

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Formula of Great Circle

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The formula of the great circle distance between two points P and Q on the surface of a sphere is given by

d = r. ?θ

Where, r is the radius of sphere and θ is the central angle between the two points

Formula of the great circle 

As per the spherical law of cosines, the central angle between the two points P and Q will be given by -

?θ = cos−1 [cosδ1 cosδ2 cos(λ1− λ2) + sinδ1 sinδ2]

Therefore, the great circle distance between two points on the earth’s surface can be calculated by the formula-

d = r cos−1 [cosδ1 cosδ2 cos(λ1− λ2) + sinδ1 sinδ2]

Where,

• r is the radius of earth = 6400 km
• δ is the latitude
• λ is the longitude

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Application of Great Circle 

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The concept of the great circle is widely applied in geographical studies, especially in the navigation of ships and aeroplanes. They are very useful in planning optimum routes. The pilots and sailors have to constantly adjust their path to match the arc of the great circle in order to remain on the optimum route i.e. the shortest distance between the origin and destination point of their journey. 

Application of Great Circle in Navigation
 

The Great Circles are also used by meteorologists for determining climate and weather conditions of various regions on earth.

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Things to Remember 

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• A great circle is created when a plane passes through the surface of the sphere while intersecting its centre.
• Great circles are the largest circles that can be drawn from the surface of a sphere.
• The plane of intersection forming the great circle always bisects the sphere into 2 equal halves.
• The shortest distance between two points on a sphere is called the great circle distance
• The diameter of a great circle is equal to the diameter of the sphere from which it is drawn.
• The centre of the great circle coincides with the centre of the sphere

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