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# Gnomonic projection

The gnomonic map projection displays great circles as straight lines.

Thus the shortest route between two locations in reality corresponds to that on the map. This is achieved by projecting, with respect to the center of the Earth (hence perpendicular to the surface), the Earth's surface onto a tangent plane. The least distortion occurs at the tangent point. Projection of even just an entire hemisphere requires an infinitely large map.

Since meridians are great circles, also they are shown as straight lines:

• if the tangent point is one of the Poles then the meridians are radial and equally spaced
• if the tangent point is on the equator then the meridians are parallel but not equally spaced
• in other cases the meridians are radially outward straight lines from a Pole, but not equally spaced

In the first case the equator is at infinity in all directions, in the second case it is a straight line perpendicular to the meridians, and in the third case it is a straight line that is perpendicular to only one meridian (which again demonstrates that the projection is not conformal).

As for all azimuthal projections, angles from the tangent point are preserved. The map distance from that point is a function r(d) of the true distance d, given by

r(d) = ctan(d/R)

where R is the radius of the Earth. The radial scale is

r′(d) = c/(2Rcos2(d/2R)

and the transverse scale

c/(2Rcos(d/2R)),

so the transverse scale increases outwardly, and the radial scale even more.

The gnomonic projection is said to be the oldest map projection, developed by Thales in the 6th century BC.