Gnomonic projection ($(\xi ,\eta )$)

The gnomonic projection sends the visible open hemisphere of a sphere to a tangent plane by central rays. Work on the unit sphere. Choose a unit vector $\mathbf{b}$ towards the tangent point and a plane $\mathbf{x}\cdot \mathbf{b}=f$, where $f>0$. Choose an orthonormal basis $\mathbf{e},\mathbf{n}$ perpendicular to $\mathbf{b}$. A point $\mathbf{s}$ with $\mathbf{s}\cdot \mathbf{b}>0$ projects toThe positivity condition ensures that the forward ray meets the chosen plane. For $f=1$ the plane is tangent to the unit sphere; changing $f$ rescales the image.

A great circle lies in a plane through the origin. Its projected arc lies in the straight line where that plane meets the image plane. This makes the gnomonic projection useful for drawing shortest spherical routes. For the northern hemisphere, take $\mathbf{b}=(0,0,1)$ and $f=1$: the map is $(x,y,z)\mapsto (x/z,y/z)$. The inverse sends $(\xi ,\eta )$ to $(\xi ,\eta ,1)/\sqrt{1+{\xi }^2+{\eta }^2}$. As a point approaches the bounding great circle $z=0$, its image escapes to infinity. The opposite hemisphere needs a separate chart; using the same formula there would identify antipodal points.