Thursday, January 7, 2016

Great circle distance

Earth is sphere and more accurately a spheroid. Calculating distance between two points on earth is different to calculating the distance between the two points on flat surface. The Circle that connects the two points with center of the sphere as the center is called great circle. The distance between the points is the length of the arc between the points on great circle.

Approximate method:
Circumference of earth at equator is 40,076km and we have 360 degrees in circle and so each degree of longitude at equator is about to 111.32km or 69 miles. The distance between two longitude lines decrease from the equator to poles.

Around 30 degrees north or south from the equator, one degree of longitude is about 96.41km. Around 45 degrees north or south of equator, one degree of longitude is 78.71km. Around 60 degrees, one degree of longitude is 55.66km. Around 75 degrees north or south of equator, one degree of longitude is 28.82km. Around 90 degrees from the equator, that will be poles and one degree of longitude is zero distance.  

However the distance between two lines of latitude stay almost same anywhere from equator to poles. Small difference is because of the shape of the earth. One degree of latitude is 110.57km at equator and 111.69km at poles.

Once we have longitudinal distance and latitude distance between two points, we can compute the distance between two points. The Pythagoras theorem can be used here.

Using cosine law:
Suppose the geographical latitude and longitude of two points on the earth are a1, a2 and b1. We can compute the central angle between them is given by the spherical law of cosines
c = arccos(sin(a1).sin(a2) + cos(a1).cos(a2).cos(|b2-b1|))
The distance d, or the arc length on a sphere of radius r formed by spherical angle c in radians  = r. c; 

Using haversine law:
Alternatively if we know the linear distance between the points as w and using sine law, length of arc on great circle can be calculated as 2*r*arcsin(w/(2.r)).

Lat Lang Story
Great circle distance using cosine law
Great circle distance using sine law
Longititude and latitude measuring

No comments: