Ellipsoid-of-Revolution to Sphere and from Sphere to Plane - Map Projections: Cartographic Information Systems - page 287

Postulate.

The spherical latitude φ should only be a function of the ellipsoidal latitude Φ : meridians of the
ellipsoid-of-revolution (lines of constant longitude) should be transformed into meridians of the
sphere (ellipses of constant longitude).

End of Postulate.

9-11 The Setup of the Mapping Equations “Ellipsoid-of-Revolution
to Plane”

λ = λ 0 + a ( Λ − Λ 0 ) , φ = f ( Φ ) . (9.1)

Λ 0 is the ellipsoidal longitude of the reference point P 0 ( Λ 0 ,Φ 0 ), an element of the ellipsoid-of-
revolution. First, let us compute the metric tensor ( first differential form ) of the ellipsoid-of-
revolution and of the sphere. Second, let us compute the curvature tensor ( second differential
form ) of the ellipsoid-of-revolution and the sphere. The mapping equations ( 9.2 )( X = Φ − 1 ( U )
versus x =( φ − 1 ( u )) form the basis of the computation of the first differential form and the second
differential form of a surface. They lead to the inverse mapping equations ( 9.3 ).

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

X
Y
Z

cos Φ cos Λ
cos Φ sin Λ
(1 − E ) 2 sin Φ

cos φ cos λ
cos φ sin λ
sin φ

x
y
z

A 1

⎣ ⎦ = ⎣ ⎦ ⎣ ⎦ = ⎣ ⎦ ,

1

versus r (9.2)

E 2 sin 2 Φ

−

U
V

= Λ = arctan YX − 1 u

v

= λ

versus 1

1 −E 2

Z
√ X 2 + Y 2

arctan

Φ φ

= arctan yx − 1

.

(9.3)

√ x 2 + y 2

x

arctan

9-12 The Metric Tensor of the Ellipsoid-of-Revolution, the First
Differential Form

First, let us here compute the first differential form of the surface of type ellipsoid-of-revolution
as follows.

3

∂X J
∂U K

∂X J
∂U L ,

G KL = G K | G L = (9.4)

J =1