Angle Between Two Points on the GlobeDate: 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/ |
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