Gall stereographic projection

Gall stereographic projection of the world. 15° graticule.

Gall stereographic projection with 1,000 km indicatrices of distortion.

TheGall stereographic projection,presented byJames Gallin 1855, is acylindrical projection.It is neitherequal-areanorconformalbut instead tries to balance the distortion inherent in any projection.

Formulae

The projection is conventionally defined as:

$x={\frac {R\lambda }{\sqrt {2}}}\,;\quad y=R\left(1+{\frac {\sqrt {2}}{2}}\right)\tan {\frac {\varphi }{2}}$

whereλis the longitude from the central meridian in degrees,φis the latitude, andRis the radius of the globe used as the model of the earth for projection. It is a perspective projection if the point of projection is allowed to vary with longitude: the point of projection being on the equator on the opposite side of the earth from the point being mapped and with the projective surface being a cylinder secant to the sphere at 45°N and 45°S.Gall called the projection "stereographic" because the spacing of theparallelsis the same as the spacing of the parallels along the centralmeridianof the equatorialstereographic projection.

The reverse projection is defined as:

$\lambda ={\frac {x{\sqrt {2}}}{R}};\quad \varphi =2\arctan {\frac {y}{R\left(1+{\frac {\sqrt {2}}{2}}\right)}}$

Braun stereographic projection

This later (1867) cylindrical projection by Carl Braun is similar, differing only in the asymmetric scaling horizontally and vertically. This yields a projection tangent to the sphere.Its formula is:

$x=R\lambda \,;\quad y=2R\tan {\frac {\varphi }{2}}$

See also

• List of map projections

Bibliography

• Snyder, John P (1993).Flattening the Earth.University of Chicago Press.ISBN978-0-226-76747-5.

Further reading

• Snyder, John P (1987).Map Projections—A Working Manual: USGS Professional Paper 1395.U.S. Government Printing Office.doi:10.3133/pp1395.

External links

• gall in proj4
• Progonos