Coordinates - Map Projections: Cartographic Information Systems - page 167

Solution (the first step).

R ( A )
A 1 √ 1 − E 2 =(1

x ) − 1 / 2 =1+ 1

2 x + 1

· 3

4 x 2 + 1

· 3 · 5

6 x 3 +O( x 4 ) ,

− (3.118)

2 · 2 · 4 ·

subject to

E 2 (1 cos 2 I sin 2 A ) , x := − − | x |≤ 1; (3.119)

E 2 1+ 1

R ( A )= A 1 √ 1

2 E 2 (1

cos 2 I sin 2 A )+

− −

6 E 6 (1 − cos 2 I sin 2 A ) 3 +O( E 8 ) .

+ 1 · 3
2 4 E 4 (1 − cos 2 I sin 2 A ) 2 + 1 · 3 · 5

(3.120)

· 2 · 4 ·

End of Solution (the first step).

Solution (the second step).

R 1 ( A )= d R

d A =

= A 1 √ 1

E 2 −

E 2 cos 2 I sin A cos A

− −

1 · 3

E 4 cos 2 I (1 cos 2 I sin 2 A )sin A cos A

− − − (3.121)

2

E 6 cos 2 I (1 − cos 2 I sin 2 A )sin A cos B − O 1 ( E 8 ) .

1 · 3 · 5
2 · 4

−

End of Solution (the second step).

Solution (the third step).

E 2 ) 1+ E 2 (1

R 2 ( A )+ R 1 ( A )= A 1 (1 cos 2 I cos 2 A )+

− −

+ E 4 (1 cos 2 I cos 2 A ) 2 +cos 4 I sin 2 A cos 2 A + − (3.122)

+ E 6 (1 − cos 2 I cos 2 A ) 3

cos 2 I sin 2 A )sin 2 A cos 2 A +O 1 ( E 8 ) . +3 cos 4 I (1 −

End of Solution (the third step).

Solution (the fourth step).

d S
d A

1
A 1 √ 1 E 2 =(1+ x ) +1 / 2 =1+ 1

1

2 · 4 x 2 + 1

· 1 · 1 · 3

2 · 4 · 6 x 3 2 x

−

−

1 · 1 · 3 · 5

2 · 4 · 6 · 8 x 4 +O + ( x 5 ) ,

− (3.123)