Computation of Distance

We have seen that there is a direct relationship between distance and the longitude or latitude. We can easily compute the aerial distance between the two points on earth if we know their coordinates accurately. For rough computations of distance, we may simply add up the difference of longitude and latitude and equate that to miles. 
Example : Calculate the distance between Delhi and Mumbai approximately.


Distance between two points on the earth's surface having longitude and latitude L1, f1 and L2, f2 respectively can be computed accurately by first computing the angular distance between the following points by the following formula :
Cos d = Sinf1.Sinf2+Cosf1.Cosf2.Cos (L1-L2) 
then computing the required linear distance by the following formula :- S = 6371p d/180 km.
where d is expressed in degrees.
(Note: The formula does not work well for very small values of d)
Example : Calculate the distance between Delhi and Mumbai, taking the following coordinates.

 

Note : - The result is accurate up to a few Km. The inaccuracy is mainly due to flattening of the earth, which has been ignored in the present formula.

Variation of Longitude and Latitude in a City

For latitudes like in India a city, which has span of 40 Km. or a distance of 25 miles, can make a difference of 25' in longitude and latitude. This is particularly so in case of Delhi and Mumbai, where the city stretches to over 40 km. diagonally. In Delhi, where the accepted coordinates 28°39' N & 77°13' E are for New Delhi Railway Station, easily makes a difference of over 25' in longitude and latitude from one end to the other. 
For example Nangal in South-West of Delhi has coordinates 28°33'N and 77°06'E, whereas Vikas Kunj in North-East coordinates of 28°45'N and 77°18'E, thus reflecting a difference of 12' in longitude and 12' in latitude, with a total difference of 24'. Similarly in Bombay Dahisar in North has coordinates 19°16'N and 72°51'E, whereas Colaba in South has coordinates 18°54'N and 72°49'E, making a difference of 22' in latitude and 2' in longitude, again making a total difference of 24'.
However small cities are normally only half or even less than half the size of Delhi or Bombay. Towns are only a few kms. in length or breadth, thus making a total difference of few minutes in all. If a centre point is chosen then the difference does not exceed more than 1' or 2' in latitude and longitude combined. For this reason for most of the towns and places, 1' accuracy in longitude or latitude is just sufficient, whereas in metropolitan cities a further breakup into small area is advisable.
To understand the total difference caused by longitude and latitude, let us convert the maximum combined difference in Delhi or Bombay (~25') into time. We find it is equivalent to 100s of time. And from a centre point it is only ±50s. i.e. less than a minute! Thus when time of birth is accurate only to a minute level, taking the centre point of even the metros for longitude or latitude is not going to add much to the inaccuracy in results.

Span of 1' of Longitude or Latitude :

Let us determine the distance represented by 1' of the longitude or latitude; that is, how much distance changes the coordinates by 1'. 
Earth's mean radius "a" = 6371 km.
Considering the earth as sphere 1degree of longitude at latitude f= p. acosf km.180.   ....{ i }.
At Delhi (f = 28° 39' N) 1° of longitude = 97.6 Km. Thus 1' of longitude at Delhi =1.626Km. and 1° of latitude = ª 1 mile p x a Km.180......{ ii }.
For all longitudes 1° of latitude = 111.2 Km. or 1' of latitude= 1.853 Km. ª 1.16 mile. For all practical purposes, in India we can consider 1 mile, making a difference of 1' in longitude or latitude or combined difference of both.
 From formula 1 and 2 it is obvious that the distance in North or South makes a variation in latitude, which is constant for all places on earth. However the longitude at least changes as much as latitude and the variation becomes more and more prominent as latitude increases. This is obvious because of the fact that equator the circumference of earth is maximum, where as it reduces as latitude increases and rate of change of circumference also increases with the increase in latitude. A table can be drawn for distance covered by 1' of longitude at different latitudes.