Geodesics - Map Projections: Cartographic Information Systems - page 738

(ii) Universal Mercator Projection (UM) :
q 1 = AΛ, q 2 = Af ( Φ ) ,

(E.72)

f ( Φ ):=lntan π

1 − e sin Φ
1+ e sin Φ

2 ,

4 + Φ

(E.73)

2

λ = 1 − e 2 sin 2 Φ
cos Φ

= 1 − e 2 sin 2 f − 1 ( q 2 /A )
cos f − 1 ( q 2 /A )

,

(E.74)

d s 2 = 1 − e 2 sin 2 f − 1 ( q 2 /A )
cos 2 f − 1 ( q 2 /A )

[(d q 1 ) 2 +(d q 2 ) 2 ] . (E.75)

( λ ( q 2 ) from series inversion of q 2 ( Φ ) leading to Φ ( q 2 ) .
See Snyder ( 1987a , p. 45).

(iii) Universal Transverse Mercator Projection (UTM) :
q 1 = ρ [ a 0 + a 10 b + a 20 b 2 + a 02 l 2 + a 30 b 3 + a 12 bl 2 + a 40 b 4 + a 22 b 2 l 2 + a 04 l 4
+ a 50 b 5 +
+ a 32 b 3 l 2 + a 14 bl 4 +O 1 ( b 6 ,l 6 )] (northern) ,

(E.76)

q 2 = ρ [ a 01 l + a 11 bl + a 21 b 2 l + a 03 l 3 + a 31 b 3 l + a 13 bl 3 + a 41 b 4 l +
+ a 23 b 2 l 3 + a 05 l 5 +O 2 ( b 6 ,l 6 )] (eastern) .

(Valid ∀ b ;= Φ Λ 0 ,ρ := 0 . 999578 (dilatation factor) .

Coecients see E. Grafarend(1994 , Table 3 . 3) .

− Φ 0 ,l := Λ −

1

λ 2 = ρ 2 [1 + c 02 l 2 + c 12 bl 2 + c 04 l 4 + c 22 b 2 l 2 + c 14 bl 4 + c 32 b 3 l 2 +O λ 2 ( b 6 ,l 6 )] ,
λ 2 = 1

ρ 2 [1 + d 02 ( q 2 ) 2 + d 12 q 1 ( q 2 ) 2 + d 22 ( q 1 ) 2 ( q 2 ) 2 +

(E.77)

+ d 32 ( q 1 ) 3 ( q 2 ) 2 + d 04 ( q 2 ) 4 + d 14 q 1 ( q 2 ) 4 +O λ 2 (( q 1 ) 6 , ( q 2 ) 6 )] ,
d s 2 = λ 2 ( q 1 ,q 2 )[(d q 1 ) 2 +(d q 2 ) 2 ] . (E.78)

( λ 2 ( q 1 ,q 2 ) from series inversion of q 1 ( l,b )and q 2 ( l,b ) leading to l ( q 1 ,q 2 )
and b ( q 1 ,q 2 ) . )

The coecients c μν and d μν are defined as follows

e 2

( η 0 =

e 2 cos 2 Φ 0 and t =tan Φ 0 ):
c 02 =(1+ η 0 )cos 2 Φ 0 ,
c 12 = − 2 t 0 (1 + 2 η 0 )cos 2 Φ 0 ,

1 −