Hammer retroazimuthal projection

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.

The full Hammer retroazimuthal projection, 15° graticule, 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.

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

${\displaystyle {\begin{aligned}x&=RK\cos \varphi _{1}\sin(\lambda -\lambda _{0})\\y&=-RK{\big (}\sin \varphi _{1}\cos \varphi -\cos \varphi _{1}\sin \varphi \cos(\lambda -\lambda _{0}){\big )}\end{aligned}}}$

where

${\displaystyle K={\frac {z}{\sin z}}}$

and

${\displaystyle \cos z=\sin \varphi _{1}\sin \varphi +\cos \varphi _{1}\cos \varphi \cos(\lambda -\lambda _{0})}$

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.