Ellipsoid-of-Revolution to Sphere and from Sphere to Plane - Map Projections: Cartographic Information Systems - page 303

sin Φ

2 E ln 1+ E sin Φ

1

(1 −E 2 sin 2 Φ ) +

sin φ = (1 − E 2 )
a

1 −E sin Φ

.

1+ 1 − E 2
2 E

ln 1+ E
1 −E

Example :
a =1 . (9.74)

Λ 1 = ar cos φ E 2 sin 2 Φ ) 1 / 2 ,

A 1 cos Φ (1

−

(9.75)

r d φ/ d Φ
A 1 (1

E 2 sin 2 Φ ) 3 / 2 ,

Λ 2 =

E 2 ) (1

−

−

A 1 (1 E 2 )

d φ
d Φ =

1
a cos φ

1
r 2

−

E 2 sin 2 Φ ) 2 cos Φ

(9.76)

(1 −

⇒

Λ 2 = Λ − 1 =

= A 1 cos Φ
a cos φ

1
r (1

E 2 sin 2 Φ ) − 1 / 2 = 1
a

A 1 cos Φ
r cos φ

1
(1 − E 2 sin 2 Φ ) 1 / 2 .

− (9.77)

we here also note the remarkable representations for sin φ (sin Φ ). Furthermore, note the remarkable
representations for the cases A 1 = r versus 4 πr 2 =4 πA 1 (1 / 2+[(1 − E 2 ) / 4 E ]ln[(1+ E ) / (1 − E )])

and Λ 1 and Λ 2 , respectively.
The final step are the standard mapping procedures of mapping the sphere to the plane. In the
following chapters, we study the mapping of the sphere to the cylinder.