Coordinates - Map Projections: Cartographic Information Systems - page 166

A 2 = A 1 (1 E 2 ) , −

A 1 (1 E 2 )=

−

E 2 ) X 2 +[(1 E 2 )cos 2 I +sin 2 I ] Y 2 +[(1 E 2 )sin 2 I +cos 2 I ] Z 2 +

=(1 − − −

+2 Y Z E 2 sin I cos I
and
A 1 (1 − E 2 ) =

(3.112)

= X 2 + Y 2 + Z 2 E 2 ( X 2 + Y 2 cos 2 I + Z 2 sin 2 I )+

+2 Y Z E 2 sin I cos I,
A 1 (1

−

E 2 )

−

=

R 2

=1 − E 2 (cos 2 A cos 2 B +sin 2 A cos 2 B cos 2 I +sin 2 B sin 2 I
− 2sin A sin B cos B sin I cos I )

(3.113)

E 2 [cos 2 A cos 2 B +(sin A cos B cos I sin B sin I ) 2 ] =1 − −

⇒
( 3.109 ) .

End of Proof.
Next,wehavetocomputethe arc length of

1

A 1 ,A 2 , i.e. that part of the oblique ecliptic equator
which ranges from the oblique quasi-spherical longitude zero to a fixed, but arbitrary value A .

E

3-44 The Arc Length of the Oblique Equator in Oblique
Quasi-Spherical Coordinates

In order to compute the length of an arc in the oblique ecliptic equator E A 1 ,A 2 in terms of oblique
quasi-spherical longitude, we are forced to represent the infinitesimal arc length by
d S = d X 2 +d Y 2 B = = R 2 ( A )+ R 1 ( A )d A, (3.114)

subject to

A 1 √ 1 − E 2

R 0 ( A, B =0):= R ( A ):= 1 , (3.115)

co s 2 I sin 2 A ) − E 2 (1 −

A 1 E 2 √ 1

E 2 cos 2 I sin A cos A

R 1 ( A, B =0):= R 1 ( A ):= 1
1!

d R ( A )
d A

−

= − , (3.116)

cos 2 I sin 2 A )] 3 / 2 [1 − E 2 (1 −

such that

S ( A )= A
A =0

R 2 ( A ∗ )+ R 1 ( A ∗ )d A ∗ .

(3.117)

In the following steps, we perform the integration. (i) Series expansion of R ( A ) according to ( 3.120 )
up to order E 6 . (ii) Series expansion of R 1 ( A )=d R/ d A accordingto( 3.121 )uptoorder E 6 . (iii)
Series expansion of R 2 ( A )+ R 1 ( A )accordingto( 3.122 )uptoorder E 6 . (iv) Series expansion of
( R 2 ( A )+ R 1 ( A )) 1 / 2 according to ( 3.125 )uptoorder E 6 .