“Ellipsoid-of-Revolution to Cone”: Polar Aspect - Map Projections: Cartographic Information Systems - page 457

⎧
⎨

1+ E sin Φ 1
1

E/ 2

Φ 2 n

tan 4 −

cn
N 1 sin Δ 1 − E sin Φ 1

Λ 1 ( Δ 1 )= Λ 2 ( Δ 2 )=1 ⇒ (19.25)

=

1+ E sin Φ 2
1 −E sin Φ 2

Φ 2 n

⎩

E/ 2

tan 4 −

cn
N 2 sin Δ 2

,

⎡ ⎤

1+ E cos Δ 2
1 −E cos Δ 2

E/ 2 n

tan Δ 2

E 2 cos 2 Δ 1 ) 1 / 2 sin Δ 2 (1 −

⎣ ⎦

E 2 cos 2 Δ 2 ) 1 / 2 =

(19.26)

1+ E cos Δ 1
1 −E cos Δ 1

E/ 2

sin Δ 1 (1 −

tan Δ 2

⇒

ln[(1 − E 2 cos 2 Δ 1 )sin 2 Δ 2 ] − ln[(1 − E 2 cos 2 Δ 2 )sin 2 Δ 1 ]
ln tan Δ 2

n = 1
2

. (19.27)

+ 2 ln (1+ E cos Δ 2 )(1 − E cos Δ 1 )

tan Δ 2

(1 − E cos Δ 2 )(1+ E cos Δ 1 )

The general form of the conformal, conic mapping takes the special form ( 19.28 ) subject to the
equidistant mapping of the parallel circle ( 19.29 ), a formula from which we derive the constant c
and finally the radial component r accordingto( 19.30 )and( 19.31 ).
r = c 1+ E cos Δ
1

n

E/ 2

tan Δ
2

, (19.28)

− E cos Δ

E 2 cos 2 Δ 1 = cn 1+ E cos Δ 1

n

E/ 2

A 1 sin Δ 1

tan Δ 1
2

√ 1

, (19.29)

1 − E cos Δ 1

−

A 1 sin Δ 1

E 2 cos 2 Δ 1 1+ E cos Δ 1 tan Δ 2 n

c = (19.30)

E/ 2

n √ 1 −

1 −E cos Δ 1

⇒

⎡ ⎤

n

1 −E cos Δ E/ 2 tan 2
1+ E cos Δ 1
1 −E cos Δ 1

1+ E cos Δ

A 1 sin Δ 1

⎣ ⎦

n √ 1

r = . (19.31)

E/ 2

− E 2 cos 2 Δ 1

tan Δ 2

19-23 Special Conic Projections of Type Equal Area

We here depart from the postulate of equal area. Fixing the integration constant, we finally arrive
at the general form of the mapping equations, namely ( 19.35 ).