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Subject: Re: how to judge if an arc of a great circle crosses a quadrangle and calculate the positions of the intersects of  on a sphere surface
Date: Sat, 13 Mar 2010 01:40:23 +0000 (UTC)
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"Matt J " <mattjacREMOVE@THISieee.spam> wrote in message <hnem4s$b39$1@fred.mathworks.com>...
> Another approach. It might boil down to Roger's, but I'm not sure.
> It assumes that the arc AB is not necessarily the shortest of the 2 arcs on the great circle between A and B, which you did not stipulate.
> ........

  I made the fundamental assumption that all great circle arcs, arc AB and the four arcs of the quadrangle, are each the shortest distance on the surface of the sphere between their respective endpoints and therefore must be great circle arcs of no more than pi radians (180 degrees).  That's crucial to the algorithm I presented.  Otherwise it doesn't work.

  Of course if arcs greater than pi are allowed and if it is specified which of two possible arcs are chosen connecting each pair of vertices, then games could be played with the signs of the s quantities to find all possible intersection points.  Without this additional information, however, the problem would not be well-defined.  Even with this information it would be possible for two arcs to intersect in two antipodal points, leaving ambiguity as to which one ought to be chosen.

Roger Stafford