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Angle Between Two Points on the Globe


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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