Ellipsoid-of-Revolution to Tangential Plane - Map Projections: Cartographic Information Systems - page 259

Corollary 8.1 (Power series (1 + x ) y , Taylor expansion).

(1 + x ) y =

=1+ 1

x =0 x 1 + 1

1! y [1 + x ] y− 1

1)[1 + x ] y− 2

x =0 x 2 +

2! y ( y

−

+ 1 2)[1 + x ] y− 3

x =0 x 3 +O( x 4 )

3! y ( y

− 1)( y − (8.32)

∀| x | < 1 ,

3
2 .

(1 + x ) y := (1 E 2 cos 2 Δ ) − 3 / 2 , x := E 2 cos 2 Δ, y := − − −

End of Corollary.

Corollary 8.2 (Power series (1 + x ) y , reformulation 1).

E 2 cos 2 Δ ) − 3 / 2 =

(1 −

=1+ 3

2 E 2 cos 2 Δ + 3

· 5

4 E 4 cos 4 Δ + 3

· 5 · 7

6 E 6 cos 6 Δ +O( E 8 ) .

(8.33)

2 · 2 · 4 ·

End of Corollary.

Corollary 8.3 (Power series (1 + x ) y , cosine powers).

cos 2 Δ = 1

2 + 1 2 cos 2 Δ,

cos 4 Δ = 3

8 + 1 2 cos 2 Δ + 1 8 cos 4 Δ,

(8.34)

16 + 15

5

32 cos 2 Δ + 3 16 cos 4 Δ + 1

cos 6 Δ =

32 cos 6 Δ.

End of Corollary.

Corollary 8.4 (Power series (1 + x ) y , reformulation 2).

E 2 cos 2 Δ ) − 3 / 2 = (1 −

2 E 2 1 2 cos 2 Δ +

=1+ 3

2 + 1

8 E 4 3 8 cos 4 Δ +

+ 15

8 + 1 2 cos 2 Δ + 1

(8.35)

16 E 6 5 32 cos 6 Δ +

+ 35

16 + 15 32 cos 2 Δ + 3 16 cos 4 Δ + 1

+O( E 8 ) .

End of Corollary.