“Sphere to Cone”: Polar Aspect - Map Projections: Cartographic Information Systems - page 436

Fig. 17.10. Mapping the sphere to a cone. Polar aspect, equal area mapping, conformal on the standard parallel
Φ = Φ 0 =45 ◦

17-232 Equidistance and Conformality on the Circle-of-Contact, Point-Like Image of the North
Pole, Compare with Fig. 17.11

Starting from the general mapping equations, in ( 17.40 ) the postulate of a point-like image of
the pole is achieved by setting r

| Φ =90 ◦ := 0, which is equivalent to assigning c =2 R 2 n − 1 .We
therefore obtain after some trigonometric conversions the general mapping equations and general
left principal stretches that are defined by ( 17.48 )and( 17.49 ). The further requirement that the
parallel circle Φ = Φ 1 shall be mapped equidistantly now determines the cone constant n =sin Φ 0 .
From the postulate ( 17.50 ), we get the value ( 17.51 ).
α
r

= ,

nΛ

√ n sin 4 −

2

(17.48) 2 R Φ

Λ 1 = 2 √ n sin 4 −

2

√ n

Φ

cos 4 −

2 ,

= (17.49)

cos Φ

Φ

Λ 2 = cos 4 −

2

Φ

√ n ,

√ n

cos 4 −

Φ 2 =1 ,

Λ 1 | Φ = Φ 1 = (17.50)

n =cos 2 π

.

Φ 1
2 4 −

(17.51)