“Sphere to Cone”: Polar Aspect - Map Projections: Cartographic Information Systems - page 435

f =

R 2

1

n sin Φ + 1

2 R 2
n 2 f 2 =

−

2 c

⇒ − sin Φ + c.

For the root to be real for all Φ , the integration constant c should fulfill the inequality c ≥ 2 R n .
The general mapping equations thus are given by ( 17.40 )or( 17.41 ), and the general left principal
stretches are given by ( 17.42 ).

=

,

α
r

nΛ

(17.40)

2 R n sin Φ + c

−

= x

y

sin Φ + c cos( nΛ )

,

2 R 2
n

− (17.41)

sin( nΛ )

n

2 R n sin Φ + c
R cos Φ

−

R cos Φ

Λ 1 = , Λ 2 =

n

. (17.42)

2 R 2
n

− sin Φ + c

17-231 Equidistance and Conformality on the Circle-of-Contact, Compare with Fig. 17.10

For the reason to map the standard parallel (circle-of-contact) Φ = Φ 0 equidistantly, we claim
that ( 17.43 ) holds, with the consequence that—together with the cone constant n =sin Φ 0 —
( 17.44 ) is immediately obtained.

n

2 R n sin Φ 0 + c
R cos Φ 0

−

Λ 1 | Φ = Φ 0 = =1 , (17.43)

c = R 2 (2 + cot 2 Φ 0 ) .

(17.44)

The mapping equations therefore are provided by ( 17.45 )or( 17.46 ). The left principal stretches
are provided by ( 17.47 ).

=

,

α
r

nΛ

R

(17.45)

2

−

n sin Φ +cot 2 Φ 0 +2

= R x

y

n sin Φ +cot 2 Φ 0 +2 cos( nΛ )

,

2

−

(17.46)

sin( nΛ )

Λ 1 = √ −

2 n sin Φ + n 2 +1
cos Φ

cos Φ

√ −

, Λ 2 =

2 n sin Φ + n 2 +1 .

(17.47)