A cartographer decides to make a map of the world using a 2-point equidistant projection.

The actual great-circle distance of any point on the map to be plotted is measured from a point on the equator at 45 degrees west longitude, and the same from 45 degrees east. These two distances are then reduced to the scale of the map. The mapping of that point is then the place on the map where the linear measures from the points representing (45 W, 0 N; 45 E, 0 N) are those reduced distances. There are, in general, two points that satisfy these conditions, so points north of the equator are plotted above the midline and points south of the equator are mapped in the bottom half of the projection.

How is the equator itself represented on the resulting map? Consider it the limiting case of non-equatorial points if you like--this might be helpful for part of the answer. If more than one shape results, specify the range of longitudes along the equator that produces each shape.

call the d the distance from (45W,0N) to (45E,0N) and f(d) (notation abuse) the distance from f(45W,0N) to f(45E,0N)

135 W to 135 E (the long way that includes 45W and 45E) will be mapped to a line segment of 3*f(d).

135W to 135E (the short way that includes 180W/E) will be an ellipse with f(45W,0N) and f(45E,0N) as foci the previous line segment as the major axis.  Each point will map to two points on the ellipse, opposite one another wrt the major axis.

This can be seen because in the specified range the shortest great circle distance from the two points completes the long part of the great circle from (45W,0N) to (45E,0N) and so the sum is a constant = 3*d.  Thus the sum of the distances in the map will be a constant 3*f(d).  An ellipse is  the set of all points where the sum of the distances from the foci = a constant.

So, an ellipse and it's major axis.

Posted by Joel on 2007-01-03 19:33:41