“Sphere to Cone”: Polar Aspect - Map Projections: Cartographic Information Systems - page 434

Fig. 17.9. Mapping the sphere to a cone. Polar aspect, conformal mapping, equidistant on two standard parallels
Φ = Φ 0 =30 ◦ and Φ = Φ 0 =60 ◦ (Lambert projection)

tan 4 − n tan 4 − n

2 2

cos nΛ
sin nΛ

= R cos Φ 2
n

cos nΛ
sin nΛ

,

Φ Φ

= R cos Φ 1
n

tan 4 −

Φ 2

tan 4 −

Φ 2

(17.37)

Λ 1 = Λ 2 =

tan 4 − n tan 4 − n

2 2

Φ Φ

= cos Φ 1
cos Φ

= cos Φ 2
cos Φ

tan 4 −

Φ 2

tan 4 −

Φ 2

. (17.38)

It is worthwhile noting that this famous map ( Lambert map , also called conical orthomorphic
mapping ) has interesting limiting forms. First, if one of the poles is selected as a single standard
parallel, the cone is a plane and a stereographic azimuthal projection is generated. If the equator
or two parallels Φ = Φ 1 and Φ = Φ 1 are chosen as the standard parallels, the cone becomes a

cylinder and the Mercator projection results.

−

17-23 Equal Area Mapping (Albers Projection)

The general mapping equations for this type of mappings are derived from the requirement that
the product of the principal stretches equals unity, i.e.

f d f = R 2
n

cos Φ d Φ

f
R =1

ff = R 2 cos Φ

nf
R cos Φ

Λ 1 Λ 2 = ⇒

n ⇒

⇓ (17.39)