Ellipsoid-of-Revolution to Sphere and from Sphere to Plane - Map Projections: Cartographic Information Systems - page 288

G 1 = G Λ := ∂ X

A 1 cos Φ

∂Λ =

E 2 sin 2 Φ ) 1 / 2 ( − sin Λ E 1 +cos Λ E 2 ) ,

(9.5)

(1 −

G 2 = G Φ := ∂ X

A 1 (1 − E 2 )

∂Φ = − E 2 sin 2 Φ ) 3 / 2 (sin Φ cos Λ E 1 +sin Φ sin Λ E 2 − cos Φ E 3 ) .

(9.6)

(1 −

A 1 denotes the semi-major axis of the e llipsoid-o f-revoluti on, A 2 den otes the semi-minor axis of

the ellipsoid-of-revolution, and E = A 1 − A 2 /A 1 = 1 A 2 /A 1 defines the first numerical

eccentricity. The basis vectors finally lead to the elements of the metric tensor.

−

A 1 cos 2 Φ

E (Gauss) := G Λ | G Λ := G ΛΛ = G 11 =

E 2 sin 2 Φ , 1 −

F (Gauss) := G Λ | G Φ := G ΛΦ = G 12 = G 21 = G ΦΛ =0 , (9.7)

A 1 (1 E 2 ) 2 −

G (Gauss) := G Φ | G Φ := G ΦΦ = G 22 =

E 2 sin 2 Φ ) 3 . (1 −

9-13 The Curvature Tensor of the Ellipsoid-of-Revolution, the Second
Differential Form

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

1

H KL = G 3 |∂ 2 X /∂U K ∂U L

= det[ G KL ] [ G K,L , G 1 , G 2 ] .

(9.8)

The second differential form is related to the determinantal form of the ellipsoid-of-revolution.
We shall compute the surface normal vector G 3 and the surface tangent vectors G 1 and G 2 .
In Box 9.1 , the various steps are collected. Subsequently, we shall collect the coordinates of the
matrix H KL which are derived from the second derivatives, see Box 9.2 . In summary, we present
the coordinates of the curvature tensor of the ellipsoid-of-revolution in ( 9.9 ).

A 1 cos 2 Φ

∂ G 1 /∂U 1

L (Gauss) := G 3 | := H ΛΛ = H 11 = −

E 2 sin 2 Φ ) 1 / 2 , (1 −

∂ G 1 /∂U 2 M (Gauss) := G 3 | := H ΛΦ = H 12 = H 21 = H ΦΛ =0 , (9.9)

E 2 )

A 1 (1 −

∂ G 2 /∂U 2 N (Gauss) := G 3 | := H ΦΦ = H 22 = −

E 2 sin 2 Φ ) 3 / 2 .

(1 −