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From: firth@sei.cmu.edu (Robert Firth)
Newsgroups: comp.graphics,sci.math
Subject: Re: simple spherical algebra question
Keywords: algebra
Message-ID: <7390@aw.sei.cmu.edu>
Date: 18 Oct 88 14:51:27 GMT
References: <1187@agora.UUCP> <1192@agora.UUCP>
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In article <1192@agora.UUCP> rickc@agora.UUCP (Rick Coates) writes:

>For those of you who asked why I wanted to do this - the most obvious
>application is to plot great circle routes on a map.  If I want to
>show the course from Sidney to Seattle, I have to incrementally draw the
>great circle route between the two, since the map projection is not
>linear (the 'straight line' great circle becomes a curved line on the 
>projection).

Please forgive me if you already know this, but there is a map
projection in which all great circles are straight lines.  It is
called the "gnomonic" projection: the hemisphere is projected onto
an infinite tangent plane from a projection point at the centre
of the sphere.

Very few published maps use this projection, since for large areas
the distortion is extreme and for small areas you don't need to
use great-circle trajectories anyway.  However, if you have a
digitised world globe somewhere, it's not too hard to write some
software to generate the projection.

Taking the centre of projection as one pole in a system of spherical
surface coordinates (theta, phi), and as the centre of the plane
polar coordinates on the projection plane (r,alpha), then

	alpha = phi
	r     = R tan(theta)

(where R is the scaled radius of the Earth).  Clearly, the equator
(theta = pi/2) is mapped onto the circle at infinity, so the projection
cannot quite cover a whole hemisphere.

Hope that helps.