Ellipsoid-of-Revolution to Tangential Plane - Map Projections: Cartographic Information Systems - page 258

Box 8.4 (Two ellipsoidal coordinate systems parameterizing the oblate ellipsoid-of-
revolution).

Oblate ellipsoid-of-revolution:

A 1 ,A 2 ,A 3 := X

+ .

X 2 + Y 2
A 1

+ Z 2

3

E ∈ R |

A 2 =1 ,A 1 >A 2

∈ R (8.28)

Ansatz 1 (surface normal coordinates):
Ansatz 2 (circle reduced coordinates):
X = A 1 cos Φ cos Λ

√ 1 −E 2 sin 2 Φ ,

X = A 1 cos Φ ∗ cos Λ,

Y = A 1 cos Φ sin Λ

√ 1 −E 2 sin 2 Φ ,

Y = A 1 cos Φ ∗ sin Λ,

(8.29)

Z = A 1 (1 − E 2 )sin Φ

√ 1 −E 2 sin 2 Φ ,

Z = A 2 sin Φ ∗ ,

subject to

A 1 = √ 1

E 2 := A 1 −

A 2 A 2

and − E 2 . (8.30)

A 1

Direct and inverse transformation of surface normal latitude
Φ to circle reduced latitude Φ ∗ :

1
1 −E 2

Z
√ X 2 + Y 2

versus tan Φ ∗ = A 1

Z
√ X 2 + Y 2

tan Φ =

A 2

1
√ 1 −E 2

Z

=

√ X 2 + Y 2 ,

versus tan Φ ∗ = √ 1

√ 1 −E 2 tan Φ ∗

1

tan Φ = − E 2 tan Φ,

(8.31)

√ 1 −E 2

√ 1 −E 2 cos 2 Φ ∗ cos Φ ∗ versus

cos Φ ∗ =

√ 1 − E 2 sin 2 Φ cos Φ,

1

cos Φ =

√ 1 √ 1 −E 2

√ 1 −E 2 sin 2 Φ sin Φ.

sin Φ =

E 2 cos 2 Φ ∗ sin Φ ∗ versus

sin Φ ∗ =

√ 1

−

In most practical cases, where we are aiming at an azimuthal projection of an equidistant type
of the ellipsoid-of-revolution representing the Earth, the planets, or other celestial bodies, a series
expansion of the meridian arc length has been a sucient approximation. Accordingly, we are
going to outline the series expansion of the meridian arc length as a function of surface normal lati-
tude Φ or its complement, the polar distance Δ . In preparing such an series expansion, we have col-
lected auxiliary formulae in Corollaries 8.1 to 8.7 .First,weexpand(1+ x ) y according to B.Taylor,
just representing the meridian arc length by x :=

E 2 cos 2 Δ, y = − − 3 / 2, and | x | > 1. Second,

E 2 cos 2 Δ ) 3 / 2 in terms of powers 1 ,E 2 cos 2 Δ, E 4 cos 4 Δ, E 6 cos 6 Δ,... we represent (1 − { } .Third,

cos 2 Δ, cos 4 Δ, cos 6 Δ,...

we transform the powers .

Fourth, an explicit version of the product sums is given in Corollaries 8.4 to 8.6 . Since the power
series are uniformly convergent , we can term-wise integrate in order to achieve the meridian arc
length in Corollary 8.7 .

{ } in terms of { 1 , cos2 Δ, cos 4 Δ, cos 6 Δ,... }