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.

In general, the map will look like an odd peanut.

With a spherical earth, the equator will be represented by a horizontal line that will stretch from 180 W to 180 E.

Assign r to represent the distance of 45 degrees. A circle of radius 2r can be plotted around 45 E that will represent the 135 E (45 W) meridian. Similarly a circle of radius 2r can be plotted around 45 W that will represent the 135 W (45 E) meridian. The North and South pole will be the intersection of these two circles.

The Prime Meridian will go vertical from the mid-point of 45 E and 45 W (O E, 0 N) until it reaches the North and South Poles.  From there it will split, one side arcing toward the equator at 180 E and the other to the equator at 180 W.

There is extreme distortion throughout the map.

Posted by Leming on 2007-01-03 12:53:59