What is the length of the shortest path along the surface of the earth that joins these two cities
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Answer:
It is well known that the shortest path between two points on the surface of the earth (assumed to be a perfect sphere of radius 6400 km) is a great circle route. To get this, draw the circle (on the surface of the earth) that goes through the two points which is centred at the centre of the earth. The shorter arc of the circle between the two points is the shortest distance. In the figure below, the solid circle represents the earth (assumed to be a sphere), with centre O. There are two points X and Y on the surface (whose positions can be specified by a latitude and longitude). The dotted line represents a circle on the surface of the earth, with centre O. The length of the arc of the circle between X and Y is the great circle distance, which is the shortest distance between the two points while travelling on the surface. A traveler wishes to accomplish a complex itinerary, going from point to point. Each point is specified by its latitude in degrees north or south, and its longitude in degrees East or West. The objective is to calculate the total distance travelled by the traveler as he goes from point to point in order, if he travels by the shortest distance between any two points.