“Sphere to Cone”: Polar Aspect - Map Projections: Cartographic Information Systems - page 437

Fig. 17.11. Mapping the sphere to a cone. Polar aspect, equal area mapping, equidistant and conformal on the
standard parallel Φ = Φ 0 =45 ◦ , point-like North Pole

The final mapping equations thus are given by ( 17.52 ), and the left principal stretches are given
by ( 17.53 ). It is easily seen that for the standard parallel Φ = Φ 1 conformality and isometry is
guaranteed.
α
r

= cos 2 4 −

, x

Φ 2 Λ =2 R sin 4 −

2

cos cos 2 4 −

Φ 2 Λ

,

Φ

cos ( 4 − Φ 2 ) sin 4 −

2

sin cos 2 4 −

Φ 2 Λ

cos 4 −

Φ 2

(17.52)

2 R Φ

y

Λ 1 = cos 4 −

Φ 2

2 , Λ 2 = cos 4 −

2

Φ

cos 4 − cos 4 −

Φ 2 .

(17.53)

Φ

17-233 Equidistance and Conformality on Two Parallels (Secant Cone, H. C. Albers), Compare
with Fig. 17.12

This famous projection which was introduced by Heinrich Christian Albers (1773-1833) in 1805
has interesting limiting forms. If one of the poles is defined to be the single standard parallel,
then the Lambert azimuthal equal area projection in the polar aspect (compare with Sect. 5-23 )
is generated: the cone becomes a plane. If, on the other hand, the equator is used as the sin-
gle standard parallel, the cylindrical equal area projection (Lambert projection, compare with
Sect. 10-23 ) is obtained. In order to derive the mapping equation, we again start from Eq.( 17.43 )
and claim that for an equidistant mapping of the standard parallel Φ = Φ 1 ,wehave