Gnomonic projection

A gnomonic map projection displays all great circles as straight lines, resulting in any straight line segment on a gnomonic map showing a geodesic, the shortest route between the segment's two endpoints. This is achieved by casting surface points of the sphere onto a tangent plane, each landing where a ray from the center of the sphere passes through the point on the surface and then on to the plane. No distortion occurs at the tangent point, but distortion increases rapidly away from it. Less than half of the sphere can be projected onto a finite map. Consequently, a rectilinear photographic lens, which is based on the gnomonic principle, cannot image more than 180 degrees.

The gnomonic projection is said to be the oldest map projection, developed by Thales in the 6th century BC. The path of the shadow-tip or light-spot in a nodus-based sundial traces out the same hyperbolae formed by parallels on a gnomonic map.

Since meridians and the equator are great circles, they are always shown as straight lines on a gnomonic map.

As with 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

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

and the scale

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

Gnomonic projections are used in seismic work because seismic waves tend to travel along great circles. They are also used by navies in plotting direction finding bearings, since radio signals travel along great circles. Meteors also travel along great circles, with the Gnomonic Atlas Brno 2000.0 being the IMO's recommended set of star charts for visual meteor observations. Aircraft and ship pilots use the projection to find the shortest route between start and destination.

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