150 3 Coordinates
Box 3.21 (Taylor series expansion of the latitude function Φ(A, B), Taylor polynomials).
∆Φ = Φ − Φ 0 =
=
∂Φ
∂A
(A 0 , B 0 ) ∆A +
∂Φ
∂B
(A 0 , B 0 ) ∆B+
+
1
2
∂
2 Φ
∂A
2 (A 0 , B 0 ) (∆A)
2 +
∂
2 Φ
∂A∂B
(A 0 , B 0 ) ∆A∆B +
1
2
∂
2 Φ
∂B
2 (A 0 , B 0 ) (∆B)
2 +
+
1
6
∂
3 Φ
∂A
3 (A 0 , B 0 ) (∆A)
3 +
1
6
∂
3 Φ
∂B
3 (A 0 , B 0 ) (∆B)
3 +
+
1
2
∂
3 Φ
∂A
2 ∂B
(A 0 , B 0 ) (∆A)
2 ∆B +
1
2
∂
3 Φ
∂A∂B
2 (A 0 , B 0 ) ∆A (∆B)
2 +
+O Φ
ˆ
(∆A)
4 , (∆B)
4 ˜
,
(3.138)
∆Λ = Λ − Λ 0 =: l , ∆Φ = Φ − Φ 0 =: b ,
∆A = A − A 0 =: α , ∆B = B − B 0 =: β .
(3.139)
Definition of partial derivatives:
b 10 :=
∂Φ
∂A
(A 0 , B 0 ) , b 01 :=
∂Φ
∂B
(A 0 , B 0 ) ,
b 20 :=
1
2
∂
2 Φ
∂A
2 (A 0 , B 0 ) , b 11 :=
∂
2 Φ
∂A∂B
(A 0 , B 0 ) , b 02 :=
1
2
∂
2 Φ
∂B
2 (A 0 , B 0 ) ,
b 30 :=
1
6
∂
3 Φ
∂A
3 (A 0 , B 0 ) , b 03 :=
1
6
∂
3 Φ
∂B
3 (A 0 , B 0 ) ,
b 21 :=
1
2
∂
3 Φ
∂A
2 ∂B
(A 0 , B 0 ) , b 12 :=
1
2
∂
3 Φ
∂A∂B
2 (A 0 , B 0 ) .
(3.140)
Taylor series, powers of Taylor series:
b = b 10 α + b 01 β + b 20 α
2 + b 11 αβ + b 02 β
2 + b 30 α
3 + b 21 α
2 β + b 12 αβ
2 + b 03 β
3 + O 1
`
α
4 , β
4 ´
,
b
2 = b
2
10 α
2 + 2b 10 b 01 αβ + b
2
01 β
2 + 2b 10 b 20 α
3 +
+2 (b 10 b 11 + b 01 b 20 ) α
2 β + 2 (b 01 b 20 + b 10 b 02 ) αβ
2 + 2b 01 b 02 β
3 + O 2
`
α
4 , β
4 ´
,
l
3 = b
3
10 α
3 + 3b
2
10 b 01 α
2 β + 3b 10 b
2
01 αβ
2 + O 3
`
α
4 , β
4 ´
.
(3.141)
Note that all the partial derivatives that are quoted in these boxes can be computed by taking
advantage of the identities d tan x = (1 + tan
2 x)dx and dx = d tan x/(1 + tan
2 x), which lead to the
recursive scheme presented in Box 3.22.
Box 3.22 (Recursive relations for the partial derivatives l ij and b ij up to order three, x ∈ {Λ, Φ).
∂x
∂A
= +
1
1 + tan
2 x
∂ tan x
∂A
,
∂
2 x
∂A
2 = +
2 tan x
(1 + tan
2 x)
2
„
∂ tan x
∂A
« 2
+
1
1 + tan
2 x
∂ tan x
∂A
,
∂
3 x
∂A
3 = −2
1 − 3 tan x
(1 + tan
2 x)
3
„
∂ tan x
∂A
« 3
−
6 tan x
(1 + tan
2 x)
2
„
∂ tan x
∂A
« „
∂
2 tan x
∂A
2
«
+
+
1
1 + tan
2 x
∂
3 tan x
∂A
3
,
∂
2 x
∂A∂B
= +
1
1 + tan
2 x
∂
2 tan x
∂A∂B
−
2 tan x
(1 + tan
2 x)
2
„
∂ tan x
∂B
« „
∂ tan x
∂A
«
.
(3.142)
Box 3.21 (Taylor series expansion of the latitude function Φ(A, B), Taylor polynomials).
∆Φ = Φ − Φ 0 =
=
∂Φ
∂A
(A 0 , B 0 ) ∆A +
∂Φ
∂B
(A 0 , B 0 ) ∆B+
+
1
2
∂
2 Φ
∂A
2 (A 0 , B 0 ) (∆A)
2 +
∂
2 Φ
∂A∂B
(A 0 , B 0 ) ∆A∆B +
1
2
∂
2 Φ
∂B
2 (A 0 , B 0 ) (∆B)
2 +
+
1
6
∂
3 Φ
∂A
3 (A 0 , B 0 ) (∆A)
3 +
1
6
∂
3 Φ
∂B
3 (A 0 , B 0 ) (∆B)
3 +
+
1
2
∂
3 Φ
∂A
2 ∂B
(A 0 , B 0 ) (∆A)
2 ∆B +
1
2
∂
3 Φ
∂A∂B
2 (A 0 , B 0 ) ∆A (∆B)
2 +
+O Φ
ˆ
(∆A)
4 , (∆B)
4 ˜
,
(3.138)
∆Λ = Λ − Λ 0 =: l , ∆Φ = Φ − Φ 0 =: b ,
∆A = A − A 0 =: α , ∆B = B − B 0 =: β .
(3.139)
Definition of partial derivatives:
b 10 :=
∂Φ
∂A
(A 0 , B 0 ) , b 01 :=
∂Φ
∂B
(A 0 , B 0 ) ,
b 20 :=
1
2
∂
2 Φ
∂A
2 (A 0 , B 0 ) , b 11 :=
∂
2 Φ
∂A∂B
(A 0 , B 0 ) , b 02 :=
1
2
∂
2 Φ
∂B
2 (A 0 , B 0 ) ,
b 30 :=
1
6
∂
3 Φ
∂A
3 (A 0 , B 0 ) , b 03 :=
1
6
∂
3 Φ
∂B
3 (A 0 , B 0 ) ,
b 21 :=
1
2
∂
3 Φ
∂A
2 ∂B
(A 0 , B 0 ) , b 12 :=
1
2
∂
3 Φ
∂A∂B
2 (A 0 , B 0 ) .
(3.140)
Taylor series, powers of Taylor series:
b = b 10 α + b 01 β + b 20 α
2 + b 11 αβ + b 02 β
2 + b 30 α
3 + b 21 α
2 β + b 12 αβ
2 + b 03 β
3 + O 1
`
α
4 , β
4 ´
,
b
2 = b
2
10 α
2 + 2b 10 b 01 αβ + b
2
01 β
2 + 2b 10 b 20 α
3 +
+2 (b 10 b 11 + b 01 b 20 ) α
2 β + 2 (b 01 b 20 + b 10 b 02 ) αβ
2 + 2b 01 b 02 β
3 + O 2
`
α
4 , β
4 ´
,
l
3 = b
3
10 α
3 + 3b
2
10 b 01 α
2 β + 3b 10 b
2
01 αβ
2 + O 3
`
α
4 , β
4 ´
.
(3.141)
Note that all the partial derivatives that are quoted in these boxes can be computed by taking
advantage of the identities d tan x = (1 + tan
2 x)dx and dx = d tan x/(1 + tan
2 x), which lead to the
recursive scheme presented in Box 3.22.
Box 3.22 (Recursive relations for the partial derivatives l ij and b ij up to order three, x ∈ {Λ, Φ).
∂x
∂A
= +
1
1 + tan
2 x
∂ tan x
∂A
,
∂
2 x
∂A
2 = +
2 tan x
(1 + tan
2 x)
2
„
∂ tan x
∂A
« 2
+
1
1 + tan
2 x
∂ tan x
∂A
,
∂
3 x
∂A
3 = −2
1 − 3 tan x
(1 + tan
2 x)
3
„
∂ tan x
∂A
« 3
−
6 tan x
(1 + tan
2 x)
2
„
∂ tan x
∂A
« „
∂
2 tan x
∂A
2
«
+
+
1
1 + tan
2 x
∂
3 tan x
∂A
3
,
∂
2 x
∂A∂B
= +
1
1 + tan
2 x
∂
2 tan x
∂A∂B
−
2 tan x
(1 + tan
2 x)
2
„
∂ tan x
∂B
« „
∂ tan x
∂A
«
.
(3.142)