Korn–Lichtenstein and d’Alembert–Euler Equations - Map Projections: Cartographic Information Systems - page 717

Tab l e D. 3 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 .Parttwo

a 20 ( b )= 3 ae 2 (1

e 2 )cos b sin b

−

,

e 2 sin 2 b ) 2 2(1 −

ae 2 (1 e 2 )

a 30 ( b )= −

−

e 2 sin 2 b ) 2 ×

2(1 −

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

× [ − −

e 2 )cos b (1+2 e 2 sin 2 b )
2(1

a 21 ( b )= −

a (1 −

,

e 2 sin 2 b ) 2 −

ae 2 (1 e 2 )sin b

a 40 ( b )= −

−

e 2 sin 2 b ) 2 ×

8cos b (1 −

15+22sin 2 b )+ e 4 sin 2 b ( 20 + 9 sin 2 b )] , × [4 + e 2 ( − −

e 2 )sin b

a 31 ( b )= −

a (1 −

e 2 sin 2 b ) 2 × 6(1 −

1+ e 2 (9 10 sin 2 b )+ e 4 sin 2 b (6 4sin 2 b )] ,

× [ − − −

4+ e 2 (3+2sin 2 b ) e 4 sin 4 b )

a 22 ( b )= a cos b sin b (

− −

× ,

e 2 sin 2 b ) 2

4(1 −

ae 2 (1 e 2 )

−

a 50 ( b )=

e 2 sin 2 b ) 1 2 × 40 cos 2 b (1 −

4+8sin 2 b + e 2 (15 128 sin 2 b + 116 sin 4 b )+

× [ − −

+ e 4 sin 4 b (180 362 sin 2 b + 164 sin 4 b )+ e 6 sin 4 b (120 136 sin 2 b +27sin 4 b )] , − −

e 2 )

− a (1 −

a 41 ( b )=

e 2 sin 2 b ) 2 × 24 cos b (1 −

36 sin 2 b )+ e 4 sin 2 b (72 60 sin 2 b )+ 1+ e 2 (9

× [ − − −

+ e 6 sin 4 b (24 8sin 2 b )] , −

a

a 32 ( b )=

e 2 sin 2 b ) 2 × 12(1 −

4+8sin 2 b + e 2 (3 16 sin 2 b +4sin 4 b )+

× [ − −

+ e 4 sin 2 b (12 10 sin 2 b +4sin 4 b ) e 6 sin 8 b ]

− −

a cos b

a 23 ( b )=

e 2 sin 2 b ) 2 × 12(1 − e 2 )(1 −

5+18sin 2 b + e 2 (1+8sin 2 b 45 sin 4 b )+ × [ − −

+ e 4 sin 2 b (2 15 sin 2 b +46sin 4 b ) e 6 sin 6 b (6 16 sin 2 b )] . − − −