Xref: utzoo comp.graphics:3364 sci.math:4647
Path: utzoo!utgpu!water!watmath!clyde!att!osu-cis!tut.cis.ohio-state.edu!mailrus!purdue!decwrl!labrea!csli!rustcat
From: rustcat@csli.STANFORD.EDU (Vallury Prabhakar)
Newsgroups: comp.graphics,sci.math
Subject: Re: simple spherical algebra question
Keywords: algebra
Message-ID: <5953@csli.STANFORD.EDU>
Date: 14 Oct 88 20:58:32 GMT
References: <1187@agora.UUCP>
Reply-To: rustcat@csli.UUCP (Vallury Prabhakar)
Organization: Stanford University
Lines: 27

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?
# 

Wouldn't this line between the two points on the surface be the
geodesic curve?  Derivations of the relevant equations may be found
in most books covering differential geometry of surfaces.  

The theoretical analysis for the mid-point might be a bit more involved.
Perhaps a good approximation might be obtained by using a parametric spline 
representation for the geodesic.  Given the tangents to the surface (and
hence the curve) at the two points and the locations of the points themselves,
one could write the parametric equation for say a cubic approximation.  Setting
the parameter to 0.5 will then yield the mid-point.  

In fact, if the curve is strictly circular, then it is possible to 
derive an a priori parametric equation so that the boundary conditions AS
WELL as the mid-point may be matched up.  

Hope this helps.  

						-- Vallury Prabhakar