“Sphere to Cone”: Polar Aspect - Map Projections: Cartographic Information Systems - page 425

Fig. 17.1. Mapping the sphere to a (tangent) cone. Polar aspect. Line-of-contact: Φ 0 =30 ◦

We know that there are two fundamental rules how to map longitudes Λ and latitudes Φ .The
angle (first polar coordinate) α = α ( Λ ) of the image p of a spherical point P ( Λ P ,Φ P ) shall only
depend on its spherical longitude Λ = Λ P . In particular, corresponding arcs on the circle-of-
contact and their images shall coincide, and this is expressed by ( 17.1 ).

sin Φ 0 = R cos Φ 0

α

R 0 Λ = r 0 α, R 0 = R cos Φ 0 ⇒ RΛ cos Φ 0 = r 0 α = R 0

sin Φ 0 α

(17.1)

⇒
α = Λ sin Φ 0 .

The term n := sin Φ 0 is called the cone constant, 0 <n< 1. For n = 0, a cylindrical, for n =1,
an azimuthal mapping is generated. The second rule concerns to the second polar coordinate r