Gnomonic projection

Great circles transform to lines via gnomonic projection

Examples of gnomonic projections

A 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. As a corollary, a rectilinear photographic lens cannot encompass more than 180 degrees for the same reason.

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

• If the tangent point is one of the Poles then the meridians are radial and equally spaced. The equator is at infinity in all directions. Other parallels are depicted as concentric circles.

• If the tangent point is on the equator then the meridians are parallel but not equally spaced. The equator is a straight line perpendicular to the meridians. Other parallels are depicted as hyperbolae.

• In other cases the meridians are radially outward straight lines from a Pole, but not equally spaced. The equator is a straight line that is perpendicular to only one meridian (which again demonstrates that the projection is not conformal).

Gnomonic projection of Earth centred 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. 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.

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, and the Gnomonic Atlas Brno 2000.0 is the IMO recommended set of star charts for visual meteor observations.

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.

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

External links

• http://www.bfi.org/node/25 Description of the Fuller Projection map from the Buckminster Fuller Institute
• http://erg.usgs.gov/isb/pubs/MapProjections/projections.html#gnomonic Explanations of projections by USGS
• http://exchange.manifold.net/manifold/manuals/6_userman/mfd50Gnomonic.htm
• http://mathworld.wolfram.com/GnomonicProjection.html
• http://members.shaw.ca/quadibloc/maps/maz0201.htm
• Table of examples and properties of all common projections, from radicalcartography.net