Geodesics - Map Projections: Cartographic Information Systems - page 750

4 ρ 2 2 d 02 q 2 +2 d 12 q 1 q 2 +2 d 22 ( q 1 ) 2 q 2 +4 d 04 ( q 1 ) 2 ,

1

g 20 =

12 ρ 3 2 d 12 q 2 +( d 12 d 02 +4 d 14 )( q 2 ) 3 +4 d 22 q 1 q 2 +6 d 32 ( q 1 ) 2 q 2 ,

1

g 30 =

(E.126)

1

12 ρ 3 2 d 02 +2 d 12 q 1 +2 d 22 ( q 1 ) 2 +( d 02 +12 d 04 )( q 2 ) 2 +2 d 32 ( q 1 ) 3 +
+(2 d 12 d 02 +12 d 14 ) q 1 ( q 2 ) 3 ,

g 03 =

g 02 = g 20 , 21 = g 03 , 12 = g 30 ,

d 02 q 2 + d 12 q 1 q 2 + d 22 ( q 1 ) 2 q 2 + 2 d 04

2 d 02 ( q 2 ) 3 ,

r 10 = 1
ρ

1

−

1

2 d 12 ( q 2 ) 2 + d 22 q 1 ( q 2 ) 2 ,

1
ρ

r 01 = −

2 ρ 2 d 12 q 2 +2 d 22 q 1 q 2 +3 d 32 ( q 1 ) 2 q 2 + 2 d 14 − 2 d 02 d 12 ( q 2 ) 3 ,

1 1

r 20 =

1

2 ρ 2 ( d 02 + d 12 q 1 + d 22 ( q 1 ) 2 +(6 d 04 −

d 22 )( q 2 ) 2 + d 32 ( q 1 ) 3 +

r 11 = (E.127)

2 d 32 ) q 1 ( q 2 ) 2 ,
r 02 = r 20 ,

+(2 d 02 d 12 +6 d 14 −

1
f 10 ,h 20 =

f 20
( f 10 ) 3 ,

h 10 = −

f 10 ( g 01 ) 2 ,h 30 = 2( f 20 ) 2

f 02 f 30

( f 10 ) 4 ,

h 02 = −

( f 10 ) 5 −

2 f 02 g 02 f 03 2 f 02 g 02 f 21

h 03 =

f 10 ( g 01 ) 4 − f 10 ( g 01 ) 3 ,h 21 =

( f 10 ) 3 ( g 01 ) 2 − ( f 10 ) 3 g 01 ,

(E.128)

2 f 20 g 02
( f 10 ) 3 ( g 01 ) 2 −

f 12
( f 10 ) 2 ( g 01 ) 2 ,

h 12 =

1
g 01 , 02 =

g 02
( g 01 ) 3 ,

k 01 = −

g 01 ( f 10 ) 2 , 03 = 2( g 20 ) 2

g 20 g 03

k 20 = −

( g 01 ) 5 − ( g 01 ) 4 ,

2 f 20 g 20
( f 10 ) 4 g 01 −

g 30
g 01 ( f 10 ) 3 , 21 =

2 g 02 g 20
( f 10 ) 2 ( g 01 ) 3 −

g 21
( f 10 ) 2 ( g 01 ) 2 ,

k 30 = (E.129)