Hammer retroazimuthal projection

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The full Hammer retroazimuthal projection, centered on the north pole.
The full Hammer retroazimuthal projection centered on Mecca, with Tissot's indicatrix of deformation. Back hemisphere has been rotated 180° to avoid overlap.

The Hammer retroazimuthal projection is a modified azimuthal proposed by Ernst Hermann Heinrich Hammer in 1910. As a retroazimuthal projection, azimuths (directions) are correct from any point to the designated center point.[1] Additionally, all distances from the center of the map are proportional to what they are on the globe. In whole-world presentation, the back and front hemispheres overlap, making the projection a non-injective function. The back hemisphere can be rotated 180° to avoid overlap, but in this case, any azimuths measured from the back hemisphere must be corrected.

Given a radius R for the projecting globe, the projection is defined as:

[math]\displaystyle{ \begin{align}x &= R K \cos \varphi_1 \sin (\lambda-\lambda_0)\\ y &= -R K \big(\sin \varphi_1 \cos \varphi - \cos \varphi_1 \sin \varphi \cos (\lambda-\lambda_0)\big)\end{align} }[/math]

where

[math]\displaystyle{ K = \frac{z}{\sin z} }[/math]

and

[math]\displaystyle{ \cos z = \sin \varphi_1 \sin \varphi + \cos \varphi_1 \cos \varphi \cos (\lambda - \lambda_0) }[/math]

The latitude and longitude of the point to be plotted are φ and λ respectively, and the center point to which all azimuths are to be correct is given as φ1 and λ0.

See also

References

  1. Snyder, John P. (1993). Flattening the Earth: Two Thousand Years of Map Projections. Chicago: University of Chicago Press. pp. 228–229. ISBN 0-226-76747-7. https://books.google.com/?id=0UzjTJ4w9yEC&pg=PA282&dq=winkel. Retrieved 2011-11-14. 

External links