Patrick Kessler wrote: >Can anyone recommend an essay on steriographic projection? In particular I >am searching for a proof that circles on the sphere are mapped onto the >equatorial plane as circles. >
As ever my response to this query is via graphical rather than numeric methods using computer scale drawing at high magnification. (3200%) As the necessary extreme example I took a small circle of about 1/4 globe diameter just north of the equator. Two diameters of the circle projected onto the equatorial plane were measured. The first on a radius from the polar axis and the second at right angles to it. This second diameter appears to be about 1.95% longer. Now comes the hard part - proving it! The apparent 'cone' of projectors (rulings) from the originating circle to the south pole cannot be a 'right cone' as the circle is not perpendicular to the axis. If the equatorial plane intersected this non-cone at the same angle as the plane of the originating circle I *think* a true circle would result but it does not in the case I have chosen. Furthermore, a circle results when a right cone is cut perpendicular to its axis so two true circles in the same 'cone' of rulings would, I think, share a common axis. Again they do not in the example I have chosen. Tony Moss P.S. For UK readers I would recommend 'Maps and Air Photographs' by G.C. Dickinson pub. Edward Arnold. There is an excellent introduction on the development of mapping and map projections.
