Geodesics - Map Projections: Cartographic Information Systems - page 748

q 2 − q 0 . The coecients f μν , g μν , r μν ,h μν ,and k μν are listed in Box E.8 .

q 1 = q 0 + f 10 u + f 20 u 2 + f 02 v 2 + f 30 u 3 + f 03 v 3 + f 21 u 2 v + f 12 uv 2 + ···,
q 2 = q 0 + g 01 v + g 20 u 2 + g 02 v 2 + g 30 u 3 + g 03 v 3 + g 21 u 2 v + g 12 uv 2 + ···,

(E.122)

α = α 0 + r 10 u + r 01 v + r 20 u 2 + r 02 v 2 + r 11 uv + ··· ,

u = t cos α 0 =
= h 10 Δq 1 + h 20 ( Δq 1 ) 2 + h 02 ( Δq 2 ) 2 + h 30 ( Δq 1 ) 3 +
+ h 03 ( Δq 2 ) 3 + h 21 ( Δq 1 ) 2 Δq 2 + h 12 Δq 1 ( Δq 2 ) 2 + ··· , (E.123)

v = t sin α 0 =
= k 01 Δq 2 + k 20 ( Δq 1 ) 2 + k 02 ( Δq 2 ) 2 + k 30 ( Δq 1 ) 3 +
+ k 03 ( Δq 2 ) 3 + k 21 ( Δq 1 ) 2 Δq 2 + k 12 Δq 1 ( Δq 2 ) 2 + ···

Box E.7 (Lie recurrence up to the third derivatives, all quantities are to be taken at Taylor
point t = 0, higher derivatives are available from the authors).

q 1 = λ cos α,
q 1 t = λu = λu 1 ,

q 2 = λ sin α,
q 2 t = λv = λu 2 ,
α = λ . 2 cos α − λ . 1 sin α,

αt = λ . 2 u − λ . 1 v,

q 1 = λ cos α λ α sin α = λλ · 1 (cos 2 α +sin 2 α ) , −

q 1 t 2 = λλ · 1 ( u 2 + v 2 ) ,
q 2 = λ sin α + λ α cos α = λλ · 2 (cos 2 α +sin 2 α ) ,
q 2 t 2 = λλ · 2 ( u 2 + v 2 ) ,

α = λ · 2 cos α

λ · 1 sin α

− − λ · 2 α sin α − λ · 1 α sin α,

− ( λλ · 12 − λ · 1 λ · 2 ) v 2 +( λλ · 22 − λλ · 11 +( λ · 1 ) 2

αt 2 =( λλ · 12 − λ · 1 λ · 2 ) u 2

( λ · 2 ) 2 ) uv,
.. q 1 = λλ · 1 (cos 2 α +sin 2 a )+ λλ · 1 (cos 2 α +sin 2 α ) ,

−

(E.124)

.. q 1 t 3 =( λ ( λ · 1 ) 2 + λ 2 λ · 11 ) u 3 +( λλ · 1 λ · 2 +
+ λ 2 λ · 12 ) u 2 v +( λ ( λ · 1 ) 2 + λ 2 λ · 11 ) uv 2 +( λλ · 1 λ · 2 + λ 2 λ · 12 ) v 3 ,