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.

Due to the earth bulging at the equator and being a distorted sphere, the shortest distance from one position on the equator to another position is not along the equator, but is along a great-circle route, either north or south of the equator.

 A line representing the equator will curve north from 45 deg W in a circle (?) back to 45 deg E, with the entire northern hemisphere contained inside the circle.

Similarly, a line representing the equator will curve south and the resulting circle (?) will contain the entire southern hemisphere. I have questioned the "circle" portion of the answer because I am not certain that the curve is a perfect circle or some form of ellipsoid.

The equitorial line between 45 W and 45 E will also curve north and south to represent the short great circle route for all points between those two points. This will leave a gap between the northern and southern hemispheres such that the two hemispheres only connect at 45 W and 45 E.

Very roughly:

   _____

/            \

|             |

\===== /

/             \            

|              |

 \ ______/

 

Edited on January 3, 2007, 9:23 am

Posted by Leming on 2007-01-03 09:20:11