“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 214

Left maximal angular distortion: Ω l =2

arcsin =2 arcsin

Λ 1 Λ 2

Λ 1 + Λ 2

− 1 sin Φ

1+sin Φ

−

. (5.88)

Parameterized inverse mapping:

tan Λ = y

x , tan Φ R

x 2 + y 2 .

(5.89)

Note that the northern gnomonic projection covers all points in the half open interval π/ 2 ≤ Φ< 0.
The point Φ = 0 moves to infinity. Accordingly, for a complete gnomonic atlas of the sphere, we
need three charts: one northern, one southern, and one equatorial chart.

Question: “What made the gnomonic projection partic-
ularly useful in marine, aerial, and space navigation?”
Answer: “It is the property that the gnomonic projection
is geodesic. Geodesics, namely great circles of the sphere,
are mapped onto a straight line—a very important charac-
teristic!”

Fig. 5.20. The gnomonic projection of the sphere

Last but not least, we present to you a nice sample of the polar gnomonic projection of the sphere
in Fig. 5.21 .