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I'm trying to understand more on projections.

enter image description here

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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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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