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I'm trying to understand more on projections.

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In this example, why must one calculate the curvature of the earth (great circle) if the map is already projected? Is there a way to draw a visibly straight line on that map which also represents a straight line on the surface of the earth? Why or Why Not?

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Part of this question is answered at gis.stackexchange.com/questions/6822/…. Also, in a comment at gis.stackexchange.com/questions/6931/…, David Cary correctly points out that there does exist a projection--the Gnomonic (known since antiquity)--that maps great circles to straight lines. Perhaps this information already covers what you want to know? –  whuber Dec 1 '11 at 20:03

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The only projection for which any straight line corresponds to a great circle is the Gnomonic projection. Additionally, any straight line drawn through the center of a map in any azimuthal projection (of which the Gnomonic is one) will be a great circle. In azimuthal projections, distances can be (but are not necessarily) preserved along lines passing through the center, but never for lines that do not pass through the center.

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There is no such thing as a "straight line" on the surface of the Earth (except over human-scale distances).

There is the curve with the shortest distance between two points, which is known as a geodesic in general and on the surface of the Earth can be approximated by a great circle.

On this particular map, which appears to be equirectangular, the only straight lines that follow a great circle are the Equator and any vertical line (i.e. following a meridian, a curve of constant longitude).

-- Andy

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