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.
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Submitted by Charlie
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Rating: 4.2500 (4 votes)
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Solution:
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Much of the equator is mapped one-to-one on a straight line segment. From 135 degrees west, eastward to 135 degrees east, the actual reduced (to scale) distances are linearly plotted: from 135 west to 45 west, the farther the place being plotted is from both defining points of the projection, the farther to the left the place appears, and similarly from 45 east to 135 east. In between 45 west and 45 east, the farther the place is from the western defining point, the closer it is to the eastern.
However, west from 135 west, all the way around to 135 east, (these are the points directly opposite the two defining points on the globe) the farther the place is from 45 west, the closer it is to 45 east, so the total of the two distances is a constant, but more than the distance between the two defining points (in particular it amounts to 90+180=270 degrees of arc), so that when laid out on the mapping paper, the mapping of that section of the equator describes an ellipse surrounding the whole map projection.
The straight line segment, from paragraph 1 above, is in fact the major axis of the ellipse. |
