“Ellipsoid-of-Revolution to Cylinder”: Oblique Aspect - Map Projections: Cartographic Information Systems - page 411

Fig. 16.3. Oblique reference frame { e 1 , e 2 , e 3 , O} with respect to the normal reference frame { e 1 , e 2 , e 3 , O}

[( x 1 ) 2 +( x 2 ) 2 ] A − 2
1

+( x 3 ) 2 A − 2
2

E 2 A 1 ,A 2 := ∈ R 3 | =1 ,A 1 ∈ R + ,A 2 ∈ R + }

along the principal axes of { x

A 1 + cos 2 i

y =1 ,

[( x 1 ) 2 +( x 2 ) 2 ] A − 1 +[( x 3 ) 2 ] A − 2 = x 2 + sin 2 i

A 2 A 1

A 2 = A 1 (1 E 2 ) −

⇒ (16.28)

x 2
A 1 +

y 2
A 1 (1 E 2 ) [(1 − E 2 )cos 2 i +sin 2 i ]= x 2

y 2
A 1 (1 E 2 ) (1 − E 2 cos 2 i )=1 . A 1 + − −

End of Proof.

In the plane {x ,y }∈{ x ∈ R

2

|Ax + By + C =0 } , we introduce circle-reduced meta-longitude

α in order to parameterize E

A 1 ,A 2 , ,namelyby( 16.29 ), illustrated by Fig. 16.4 .
x = A 1 cos α = A 1 sin α ∗ , α ∗ = π

1

2 −

α,

(16.29)

y = A 2 sin α = A 2 cos α ∗ , α = π α ∗ .

2 −

In terms of circle-reduced metalongitude α or of circular reduced meta-pole distance α ∗ = π/ 2 −α ,
we are able to represent the arc length of E

1

A 1 ,A 2 , as an elliptic integral of the second kind.

Corollary 16.2 (Arc length of E

1
A 1 ,A 2

).

1
A 1

The arc length s ( α )of can be represented by ( 16.30 ) with respect to the elliptic integral

of the second kind E ( · ; E ) and the first relative eccentricity E := (1 − A 2 /A 1 ) 1 / 2 .Aseries
expansion of s ( α )uptoorder E 1 / 2 is provided by ( 16.31 ).

E

,A 2