Korn–Lichtenstein and d’Alembert–Euler Equations - Map Projections: Cartographic Information Systems - page 716

Tab l e D. 2 E a,b , the transverse Mercator projection, the series expansions x ( l,b )and y ( l,b ) based upon x ( q )=

r =0

a r q r .Partone

q ( b )= r 1 =1 c r 1 b r 1 ,

q 2 ( b )= r 1 ,r 2

r =2

c r 1 c r 2 b r 1 b r 2 = ∞

( cc ) r b r ,

q 3 ( b )=
r 1 ,r 2 ,r 3

r =3

c r 1 c r 2 c r 3 b r 1 b r 2 b r 3 = ∞ ( ccc ) r b r

etc.

e 2 ) a (1 −

a 10 ( b )= ,

3
2

e 2 sin 2 b ) (1 −

a cos b

a 01 ( b )= 1 ,

e 2 sin 2 b −

− e 2 )sin b

a 11 ( b )= − a (1 ,

3
2

e 2 sin 2 b )

(1 −

a cos b sin b
2 1

a 02 ( b )= ,

e 2 sin 2 b

−

2si n 2 b + e 2 sin 4 b )

a 03 ( b )= a cos b (1

−

e 2 ) 1

,

e 2 sin 2 b

− −

6(1

2sin 2 b + e 2 sin 4 b )

a 12 ( b )= a (1

−

,

3
2

e 2 sin 2 b )

2(1 −

a sin b

a 13 ( b )= ×

3
2

e 2 sin 2 b ) e 2 )(1 6(1 − −

5+6sin 2 b + e 2 (1+6sin 2 b 9sin 4 b )+ e 4 sin 4 b ( 3+4sin 2 ) b ], × [ − − −

a cos b sin b

a 04 ( b )=

e 2 ) 2 1 e 2 sin 2 b × 24(1 − −

6sin 2 b e 2 (1+6sin 2 b 9sin 4 b )+ e 4 sin 4 b (3 4sin 2 ) b ],

× [5 − − − −

a

a 14 ( b )= ×

3
2

e 2 sin 2 b )

24(1 − e 2 ) 2 (1 −

28 sin 2 b +24sin 4 b + e 2 ( 16 sin 2 b +86sin 4 b 72 sin 6 b )]+ × [5 − − 1 − −

+ e 4 sin 4 b (26 100 sin 2 b +77sin 4 b )+ e 6 sin 6 b ( 12 + 39 sin 2 b 28 sin 4 b )] , − − −

− a cos b

a 05 ( b )=

e 2 ) 3 1 e 2 sin 2 b × 120(1 − −

5+28sin 2 b 24 sin 4 b + e 2 (1+16sin 2 b 86 sin 4 b +72sin 6 b )]+ × [ − − −

+ e 4 sin 4 b ( 26 + 100 sin 2 b 77 sin 4 b )+ e 6 sin 6 b (12 39 sin 2 b +28sin 4 b )] . − − −