“Sphere to Cone”: Polar Aspect - Map Projections: Cartographic Information Systems - page 431

, c = R 2 −

Φ 1 cos Φ 2 − 2 − Φ 2 cos Φ 1

sin Φ 0 = n = cos Φ 1

− cos Φ 2

, (17.18)

Φ 2 − Φ 1 cos Φ 1 − cos Φ 2

=

,

α
r

cos Φ 1 − cos Φ 2
Φ 2 −Φ 1

Λ

R

(17.19)

Φ + Φ 1 cos Φ 2 − Φ 2 cos Φ 1
cos Φ 2 − cos Φ 1

−

⎡
⎣

cos cos Φ 1 − cos Φ 2 Λ

⎤

x
y

= R

−Φ + Φ 1 cos Φ 2 − Φ 2 cos Φ 1
cos Φ 2

Φ 2 −Φ 1

sin cos Φ 1 − cos Φ 2
Φ 2 −Φ 1

Λ

⎦ ,

(17.20)

− cos Φ 1

Λ 1 = Φ 2 cos Φ 1

− Φ 1 cos Φ 2 + Φ (cos Φ 2 − cos Φ 1 )

, Λ 2 =1 . (17.21)

( Φ 2 − Φ 1 )cos Φ

Fig. 17.7. Mapping the sphere to a cone. Polar aspect, equidistant mapping of the set of meridians, equidistant
and conformal on two parallels Φ = Φ 1 =0 ◦ and Φ = Φ 2 =60 ◦ (de L'Isle projection)

17-22 Conformal Mapping (Lambert Projection)

The general mapping equations for this type of mappings are derived from the identity ( 17.22 ). The
mapping equations are obtained as ( 17.25 ). The left principal stretches are obtained as ( 17.26 ).
Λ 1 = C 11

R cos Φ = Λ 2 = C 22

G 22 = f

nf

G 11 =

(17.22)

R

⇓

d f
f

= n

f
f

n
cos Φ ⇒

d Φ
cos Φ

= (17.23)