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### Angle Between Two Points on the Globe

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Date: 7/17/96 at 17:20:31
From: Jason Shelton
Subject: Angle Between Two Points on Globe

You have a page at your site telling how to determine the distance
between 2 cities by using their longitude and latitude points.

I was wondering how the solution would change if radian measure had to
be used.
```

```
Date: 7/17/96 at 19:9:1
From: Doctor Anthony
Subject: Re: Angle Between Two Points on Globe

This can be done very easily by using the scalar product of two
vectors to find the angle between those vectors. Let the vectors be OA
and OB, where A and B are the two points on the surface of the earth
and O is the centre of the earth. The scalar product gives

OA*OB*cos(AOB)  = R^2*cos(AOB)

where R = radius of the earth.  Having found angle AOB the distance
between the points is R*(AOB) with AOB in radians.

To find the scalar product we need the coordinates of the two points.
Set up a three-dimensional coordinate system with the x-axis in the
longitudinal plane of OA, the xy plane containing the equator, and the
z-axis along the earth's axis. With this system, the coordinates of A
will be

Rcos(latA), 0, Rsin(latA)

and the coordinates of B will be

Rcos(latB)cos(lonB-lonA),Rcos(latB)sin(lonB-lonA),Rsin(latB)

The scalar product is given by

xA*xB + yA*yB + zA*zB
= R^2cos(latA)cos(latB)cos(lonB-lonA)+ R^2sin(latA)sin(latB)

Dividing out R^2 will give cos(AOB)

cos(AOB)=cos(latA)cos(latB)cos(lonB-lonA)+sin(latA)sin(latB)

This gives AOB, and the great circle distance between A and
B will be

R*(AOB)   with AOB in radians.

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
College Higher-Dimensional Geometry
High School Higher-Dimensional Geometry

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