Ellipsoid-of-Revolution to Sphere and from Sphere to Plane - Map Projections: Cartographic Information Systems - page 290

E 2 )(1 + 2 E 2 sin 2 Φ ) A 1 (1 −

= −

E 2 sin 2 Φ ) 5 / 2
cos 2 Φ +sin 2 Φ 1+2 E 2 sin 2 Φ

(1 −

=

3 E 2
1+2 E 2 sin 2 Φ

−

(9.14)

E 2 )
(1 − E 2 sin 2 Φ ) 5 / 2
(cos 2 Φ +2 E 2 sin 2 Φ cos 2 Φ +sin 2 Φ +2 E 2 sin 4 Φ − 3 E 2 sin 2 Φ )

A 1 (1 −

= −

E 2 )

A 1 (1 −

E 2 (3 sin 2 Φ 2sin 2 Φ cos 2 Φ 2sin 4 Φ )] =

= −

E 2 sin 2 Φ ) 5 / 2 [1

− − −

(1 −

E 2 ) A 1 (1 −

= −

E 2 sin 2 Φ ) 5 / 2 (1 −

E 2 (3 sin 2 Φ 2sin 2 Φ cos 2 Φ 2sin 2 Φ +2sin 2 Φ cos 2 Φ )] = [1 − − −

A 1 (1 − E 2 ) A 1 (1 − E 2 )

E 2 sin 2 Φ ) 5 / 2 (1 − E 2 sin 2 Φ )= −

= −

E 2 sin 2 Φ ) 3 / 2 . (1 − (1 −

9-14 The Metric Tensor of the Sphere, the First Differential Form

Third, we compute the first differential form of the surface of type sphere .In( 9.15 )and( 9.16 ), r
is the radius of the sphere. The basis vectors finally lead to the elements of the spherical metric
tensor .

3

∂x j
∂u k

∂x j
∂u l ,

g kl = g k | g l =

j =1

g 1 = g λ := ∂ x

∂λ = r cos φ ( − sin λe 1 +cos λe 2 ) ,

(9.15)

g 2 = g φ := ∂ x

∂φ =

− r (sin φ cos λe 1 +sin φ sin λe 2 − cos φe 3 ) ,

e (Gauss) := g λ | g λ := g λλ = g 11 = r 2 cos 2 φ,

f (Gauss) := g λ | g φ := g λφ = g 12 = g 21 = g φλ =0 , (9.16)

:= g φφ = g 22 = r 2 .

g (Gauss) := g φ | g φ