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From: jdchrist@watcgl.waterloo.edu (Dan Christensen)
Newsgroups: comp.graphics,sci.math
Subject: Re: simple spherical algebra question
Keywords: algebra
Message-ID: <6337@watcgl.waterloo.edu>
Date: 15 Oct 88 20:26:44 GMT
References: <1187@agora.UUCP>
Reply-To: jdchrist@watcgl.waterloo.edu (Dan Christensen)
Organization: U. of Waterloo, Ontario
Lines: 34

In article <1187@agora.UUCP> rickc@agora.UUCP (Rick Coates) writes:
>
>
>Here is a straightforward question that I have not been able to find
>in my reference books:
>
>	Given two points on a sphere, what is the equation of the
>	line (great circle) between them?  What is the midpoint?

The equation for the great circle can be found using the technique
described in an earlier posting.

To find the midpoint, use the following method.

Assume that the sphere is centered at the origin, and has radius r.
(You can translate your sphere to this state and then
undo the translation the resulting point.)

Let's say the two points are (x1, y1, z1) and (x2, y2, z2).
The midpoint of the straight line joining these points is
p = ( (x1+x2)/2, (y1+y2)/2, (z1+z2)/2 ).

Then just scale this point to the surface of the sphere.
The midpoint of the great circle is then

    r
   --- * p, where |p| is the length of p, sqrt( x^2 + y^2 + z^2 ). 
   |p|

Hope this helps.

Dan Christensen
jdchrist@watcgl.uwaterloo.ca