Coordinates - Map Projections: Cartographic Information Systems - page 171

l 2 = l 10 α 2 +2 l 10 l 01 αβ + l 01 β 2 +2 l 10 l 20 α 3 + (3.137)

+2( l 10 l 11 + l 01 l 20 ) α 2 β +2( l 01 l 20 + l 10 l 02 ) αβ 2 +2 l 01 l 02 β 3 +O 2 ( α 4 ,β 4 ) ,
l 3 = l 10 α 3 +3 l 10 l 01 α 2 β +3 l 10 l 01 αβ 2 +O 3 ( α 4 ,β 4 ) .

Box 3.21 (Taylor series expansion of the latitude function Φ ( A, B ), Taylor polynomials).

ΔΦ = Φ − Φ 0 =

= ∂Φ

∂A ( A 0 ,B 0 ) ΔA + ∂Φ

∂B ( A 0 ,B 0 ) ΔB +

∂ 2 Φ ∂ 2 Φ ∂ 2 Φ

+ 1
2 ∂A∂B ( A 0 ,B 0 ) ΔAΔB + 1 ∂A 2 ( A 0 ,B 0 )( ΔA ) 2 + ∂B 2 ( A 0 ,B 0 )( ΔB ) 2 + 2

+ 1
6

∂ 3 Φ

∂A 3 ( A 0 ,B 0 )( ΔA ) 3 + 1

∂ 3 Φ

∂B 3 ( A 0 ,B 0 )( ΔB ) 3 +

(3.138)

6

∂ 3 Φ ∂ 3 Φ

+ 1
2 ∂A 2 ∂B ( A 0 ,B 0 )( ΔA ) 2 ΔB + 1 ∂A∂B 2 ( A 0 ,B 0 ) ΔA ( ΔB ) 2 +

+O Φ [( ΔA ) 4 , ( ΔB ) 4 ] ,

2

ΔΛ = Λ − Λ 0 =: l, ΔΦ = Φ − Φ 0 =: b,

ΔA = A − A 0 =: α, ΔB = 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
9

∂ 2 Φ
∂A 2 ( A 0 ,B 0 ) ,b 11 := ∂A∂B ( A 0 ,B 0 ) ,b 02 := 1

∂ 2 Φ ∂ 2 Φ

∂B 2 ( A 0 ,B 0 ) , 2

∂ 3 Φ ∂ 3 Φ

∂B 3 ( A 0 ,B 0 ) ,

b 30 := 1

∂A 3 ( A 0 ,B 0 ) ,b 03 := 1

(3.140)

6 6

∂ 3 Φ ∂ 3 Φ

∂A∂B 2 ( A 0 ,B 0 ) .

b 21 := 1
2 ∂A 2 ∂B ( Λ 0 ,B 0 ) ,b 12 := 1 2

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 10 α 2 +2 b 10 b 01 αβ + b 01 β 2 +2 b 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 +2 b 01 b 02 β 3 +O 2 ( α 4 ,β 4 ) , (3.141)

l 3 = b 10 α 3 +3 b 10 b 01 α 2 β +3 b 10 b 01 αβ 2 +O 3 ( α 4 ,β 4 ) .