Ellipsoid-of-Revolution to Sphere and from Sphere to Plane - Map Projections: Cartographic Information Systems - page 299

The following chains of calculations, on the one hand, supply us with proofs of the above relations,
and on the other hand, supply us with additional relations needed to understand the above
relations.

Proof (first postulate).

First postulate ( Λ 0 =1):

Λ 0 = ar cos φ 0 r = N 0 cos Φ 0

N 0 cos Φ 0 ⇒ a cos φ 0 .

(9.56)

End of Proof.

Proof (second postulate).

Second postulate( Λ 0 =0 (ln Λ ) 0 = Λ 0 /Λ 0 ):

⇔

ln Λ =ln ar +lncos φ − ln[ N ( Φ )cos Φ ] ,

N ( Φ )cos Φ

dln Λ
d φ

sin φ
cos φ −

N ( Φ )sin Φ
N ( Φ )cos Φ

− d Φ

d φ ,

= −

E 2 sin 2 Φ ) 1 / 2 ,N ( Φ )= A 1 E 2 sin Φ cos Φ

A 1

N ( Φ )=

E 2 sin 2 Φ ) 3 / 2 , (1 − (1 −

E 2 sin 2 Φ
1

d Φ

d φ = 1

−

a cos φ = N ( Φ )

cos Φ cos Φ

a cos φ

(9.57)

− E 2 M ( Φ )

⇒

dln Λ
d φ

sin φ
cos φ +

sin Φ sin Φ

a cos φ

= −

a cos φ = − tan φ +

⇒

(ln Λ ) ( φ 0 ,Φ 0 )=0

⇒ a sin φ 0 =sin Φ 0 .

End of Proof.

Proof (third postulate).

Third postulate( Λ 0 =0 (ln Λ ) 0 =0):

⇔

a cos φ cos Φ d d φ + a sin φ sin Φ
a 2 cos 2 φ

d 2 ln Λ
d φ 2

1
cos 2 φ +

= − =