8/10/2019 haversine

1/4

Distance between Points on the Earths Surface

Abstract

During a casual conversation with one of my students, he asked me how one could go about computing the

distance between two p oints on the surface of the Earth, in terms of their respective latitudes and longitudes.

This is an interesting exercise in spherical coordinates, and relates to the so-called haversine.

Spherical coordinates

z=Rsin!

y=Rcos!sin "x=Rcos!cos "

R

z

y

x

!

"

Figure 1: Spherical Coordinates

The calculation of the distance be-tween two points on the surface of theEarth proceeds in two stages: (1) to

compute the straight-line Euclideandistance these two points (obtained byburrowing through the Earth), and (2)to convert this distance to one mea-sured along the surface of the Earth.Figure 1 depicts the spherical coor-

dinates we shall use.1

We orient thiscoordinate system so that

(i) The origin is at the Earths center;

(ii) Thex-axis passes through the Prime Meridian (0 longitude);

(iii) The xy-plane contains the Earths equator (and so the positive z-axis willpass through the North Pole)

Note that the angle is the measurement of lattitude, and the angle is themeasurement of longitude, where0

8/10/2019 haversine

2/4

Thus, we assume that we are given two points P1 and P2 determined by theirrespective lattitude-longitude pairs: P1(1, 1), P2(2, 2). In cartesian coordi-nates we haveP1=P1(x1, y1, z1)andP2=P2(x2, y2, z2), wherex, y, andzaredetermined by the spherical coordinates through the familiar equations:

x = R cos cos y = R cos sin

z = R sin

The Euclidean distanced betweenP1andP2is given by the three-dimensionalPythagorean theorem:

d2 = (x1

x2)2 + (y1

y2)

2 + (z1

z2)2.

The bulk of our work will be in computing this distance in terms of the sphericalcoordinates. Converting the cartesian coordinates to spherical coordinates, weget

d2/R2 = (cos 1cos 1 cos 2cos 2)2+(cos 1sin 1 cos 2sin 2)2+(sin 1

sin 2)

2

= cos2 1cos2 1 2cos 1cos 1cos 2cos 2+ cos2 2cos2 2+cos2 1sin

2 1 2cos 1sin 1cos 2sin 2+ cos2 2sin2 2+sin2 1 2sin 1sin 2+ sin2 2

= 2 2cos 1cos 2cos(1 2) 2sin 1sin 2

8/10/2019 haversine

3/4

P2

P1

D

!

R

d/2

d/2

R

Figure 2: Arc Length

Next in order to compute the distance Dalong the surface of the Earth, we need onlyanalyze Figure 2 in detail. Notice that D is thearc length along the indicated sector. From

sin(/2) =

d

2R,we get

sin = 2 sin(/2) cos(/2)

= d

R

1

d

2R

2

= d

2R24R2 d2.

Therefore, in terms ofd and R, the distanceDis given by

D=R = R sin1

d

2R2

4R2 d2

,

which, in principle, concludes this narrative.

The distance is often represented in terms of the so-called haversinefunction,defined by

haversin A = sin2A

2

=

1

cosA

2 .

This says that cos A= 1 2 haversin A.

Returning to the formula fordabove, we continue, feeding in the new notation:

d2/R2 = 2 2cos 1cos 2cos(1 2) 2sin 1sin 2= 2 2cos 1cos 2[1 2 haversin (1 2)] 2sin 1sin 2= 2 2cos(1 2) + 4 cos 1cos 2haversin (1 2)

= 4 haversin (1 2) + 4 cos 1cos 2haversin (1 2)Which says that

d

2R

2

= haversin (1 2) + cos 1cos 2haversin (1 2),

8/10/2019 haversine

4/4

and so

haversin= haversin (1 2) + cos 1cos 2haversin (1 2),

Finally, we have (refer to Figure 2)

d2R

2= haversin (12)+cos 1cos 2haversin (12), sin2(/2) = haversin ,

meaning that

D=R = 2R sin1

haversin

.