From: Edward A Williams <72347.1516@compuserve.com>
Newsgroups: comp.infosystems.gis,alt.math,sci.math
Subject: Re: Sphere-line algorithm wanted
Date: Sat, 27 Dec 1997 12:37:30 -0800

Eric Buddington wrote:

> Context:
>
> I'm building an algorithm that takes a bunch of points and returns a
> bounding box that encompasses all the points and any lines that might
> be drawn between them.
>
> The sides of the bounding box follow lines of latitude and longitude,
> so the north and south edges aren't straight (I locally define
> 'straight' to mean 'following the shortest surface distance between
> two points).
>
> I can easily get such a box that encompasses all the points, but
> straight lines can go outside of the box along the north or south
> edges, if they are concave.
>
> Question:
>
> Given two points at the same latitude in the northern hemisphere, what
> is the northernmost point reached by a straight (see definition above)
> line between them? An algorithm that gives a high (too far north)
> answer is acceptable. Speed is important.
>
> Assume the earth is spherical (ellipsoidal geometry gets kind of
> hairy).
>
> Ideas and partial answers may also be useful.
>
> Eric
> ebuddington@wesleyan.edu

  http://www.best.com/~williams/avform.htm#Clairaut

  has Clairaut's formula which answers your question.  There is an
ellipsoidal variant of the same formula if you really need it.

                                      Ed