Ellipsoid-of-Revolution to Sphere and from Sphere to Plane - Map Projections: Cartographic Information Systems - page 300

1 − ,

1
cos 2 φ

1
a sin φ sin Φ −

1

a cos φ cos Φ d Φ

= −

d φ

d 2 ln Λ
d φ 2

( φ 0 ,Φ 0 ) = 0 (9.58)

⇒

1 − E 2 sin 2 Φ 0
1

a cos φ 0 =

cos Φ 0 .

− E 2

End of Proof.

Proof (lemma relations).

Intermediate results:

r = N 0 cos Φ 0

a cos φ 0 ,

(9.59)

a sin φ 0 =sin Φ 0 ,

(9.60)

1 − E 2 sin 2 Φ 0
1

cos Φ 0 = N 0 a cos φ 0 = a 1 sin 2 φ 0 =

−

M 0 cos Φ 0 .

(9.61)

− E 2

Action item: combine ( 9.59 )and( 9.61 ) and find

A 1 √ 1

1 − E 2 sin 2 Φ 0 = M 0 N 0 .

r = A 1 cos Φ 0
a cos φ 0

1 − E 2

1

= (9.62)

E 2 sin 2 Φ 0

−

Action item: combine ( 9.60 )and( 9.61 ) and find

1 − E 2 sin 2 Φ 0 tan Φ 0 = M 0

1 − E 2

tan φ 0 =

N 0 tan Φ 0 .

(9.63)

Action item: combine ( 9.60 )and( 9.61 ) and find
a = sin 2 Φ 0 + N 0

M 0 cos 2 Φ 0 =cos Φ 0 N 0

M 0 tan 2 Φ 0 .

(9.64)

(If N 0 /M 0 =1 , then a =1 . )

Solve the mapping equations at the initial fundamental point P 0 ( Λ 0 ,Φ 0 ) with respect to the
integration constant c and find