= Gnomonic projection
{title2=$(\xi,\eta)$}
{wiki}
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 to
$$
\mathbf x=\frac{f\mathbf s}{\mathbf s\cdot\mathbf b},\qquad
(\xi,\eta)=\frac{f(\mathbf s\cdot\mathbf e,\mathbf s\cdot\mathbf n)}{\mathbf s\cdot\mathbf b}.
$$
The 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>.
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