Hammer Aitov projection

Distortions of the Hammer-Aitov projection illustrated with the Tissot indicatrix

The Hammer-Aitov projection (also called Hammer-Aitoff projection or just Hammer projection ) is an area map projection proposed by Ernst Hammer in 1892 , which shows the entire surface of the earth as an ellipse. It is based on the Aitov projection by Dawid Aitow , but in contrast to that, it is true to area instead of length . Hammer used a Lambertian azimuthal projection instead of the equidistant azimuthal projection .

The equator and central meridian are shown true to scale and as a straight line, but the greater the distance from them, the greater the distortion. Usually the prime meridian is the central meridian .

Other longitudes and latitudes are shown as curves. The meridian opposite the central meridian forms the outer edge of the map.

The distortions in the polar regions are not as strong as with the similar looking Mollweide projection .

Formulas

{\ displaystyle x = \ mathrm {laea} _ {x} \ left ({\ frac {\ lambda} {2}}, \ phi \ right)}

{\ displaystyle y = {\ frac {1} {2}} \ mathrm {laea} _ {y} \ left ({\ frac {\ lambda} {2}}, \ phi \ right)}

where and are the x and y components of the azimuthal Lambert equal area projection. Written out: ${\ displaystyle \ mathrm {laea} _ {x}}$ ${\ displaystyle \ mathrm {laea} _ {y}}$

{\ displaystyle x = {\ frac {2 \ cdot {\ sqrt {2}} \ cdot \ cos (\ phi) \ cdot \ sin \ left ({\ frac {\ lambda} {2}} \ right)} { \ sqrt {1+ \ cos (\ phi) \ cdot \ cos \ left ({\ frac {\ lambda} {2}} \ right)}}}}

{\ displaystyle y = {\ frac {{\ sqrt {2}} \ cdot \ sin (\ phi)} {\ sqrt {1+ \ cos \ phi \ cdot \ cos \ left ({\ frac {\ lambda} { 2}} \ right)}}}}

The inverse of the projection is determined by an intermediate variable:

{\ displaystyle z: = {\ sqrt {1- \ left ({\ tfrac {1} {4}} x \ right) ^ {2} - \ left ({\ tfrac {1} {2}} y \ right ) ^ {2}}}}

Longitude and latitude can then be calculated as follows:

{\ displaystyle \ lambda = 2 \ cdot \ arctan \ left ({\ frac {z \ cdot x} {2 \ cdot (2z ^ {2} -1)}} \ right)}

{\ displaystyle \ phi = \ arcsin (z \ cdot y)}

where is the longitude and the latitude . The mapping space is in the area and . The area results from the area equation of the resulting ellipse ${\ displaystyle \ lambda}$ ${\ displaystyle \ phi}$ ${\ displaystyle x \ in \ left [-2 {\ sqrt {2}} \ ,, \ +2 {\ sqrt {2}} \ right]}$ ${\ displaystyle y \ in \ left [- {\ sqrt {2}} \ ,, \ + {\ sqrt {2}} \ right]}$ ${\ displaystyle A = a \ cdot b \ cdot \ pi}$

${\ displaystyle {\ sqrt {2}} \ cdot 2 \ cdot {\ sqrt {2}} \ cdot \ pi = 4 \ cdot \ pi}$

as the imaging surface of the unit sphere. This corresponds to the result of the spherical surface equation with r = 1. ${\ displaystyle A_ {O} = 4 \ cdot \ pi \ cdot r ^ {2}}$

To get real metric sizes, the x and y values have to be multiplied by the radius of the earth .

Web links

Commons : Hammer Aitov Projection - collection of images, videos and audio files

proof

Flattening the Earth: Two Thousand Years of Map Projections. John P. Snyder, 1993, ISBN 0-226-76747-7 , pp. 130-133.
Eric W. Weisstein: Hammer-Aitoff Equal-Area Projection. From MathWorld - A Wolfram Web Resource.
The Aitoff-Wagner Projection , in John Savard: Map Projections .