Gnomonic projection

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Great circles transform to lines via gnomonic projection
Examples of gnomonic projections

The gnomonic map projection displays all 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. Less than half of the sphere can be projected onto a finite map.

Since Meridians and the Equator are great circles, they are always shown as straight lines.

Gnomonic projection of Earth centered on the geographic North Pole

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) = R\,\tan (d/R)

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

 r'(d) = \frac{1}{\cos^2(d/R)}

and the transverse scale

 \frac{1}{\cos(d/R)}

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.

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.

[edit] History

In 1946 Buckminster Fuller patented a projection method similar to the Gnomonic Projection in his cuboctahedral version of the Dymaxion map. The 1954 icosahedral version he published under the title of AirOcean World Map, and this is the version most commonly referred to today.

[edit] External links


[edit] References

Snyder, John P. (1987). Map Projections - A Working Manual. U.S. Geological Survey Professional Paper 1395. United States Government Printing Office, Washington, D.C.  This paper can be downloaded from USGS pages

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