Coordinates - Map Projections: Cartographic Information Systems - page 170

Corollary 3.9 (The change from one chart to another chart: cha-cha-cha, the oblique quasi-
spherical longitude/latitude versus the surface normal ellipsoidal longitude/latitude).

Given the longitude of the ascending node Ω as well as the inclination I of the oblique equatorial
plane, then the transformation of oblique quasi-spherical longitude/latitude in surface normal
ellipsoidal longitude/latitude is represented by ( 3.133 ).

End of Corollary.

2
A 1 ,A 2

Next, let us assume that we know already a point { Λ 0 ,Φ 0 } , correspondingly { A 0 ,B 0 } ,in E .

Relative to such a fixed point, we are able to find the coordinates { Λ, Φ } , correspondingly { A, B } ,

close to { Λ 0 ,Φ 0 } , correspondingly { A 0 ,B 0 } , by a Taylor series expansion of the type presented in

Boxes 3.20 and 3.21 .

Box 3.20 (Taylor series expansion of the longitude function Λ ( A, B ), Taylor polynomials).

ΔΛ = Λ − Λ 0 =

= ∂Λ

∂A ( A 0 ,B 0 ) ΔA + ∂Λ ∂B ( A 0 ,B 0 ) ΔB +

∂ 2 Λ ∂ 2 Λ ∂ 2 Λ

+ 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

∂ 3 Λ ∂ 3 Λ

+ 1
6 ∂Λ 3 ( A 0 ,B 0 )( ΔA ) 3 + 1 ∂B 3 ( A 0 ,B 0 )( ΔB ) 3 +

(3.134)

6

∂ 3 Λ ∂ 3 Λ

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

∂A∂B 2 ( A 0 ,B 0 ) ΔA ( ΔB ) 2 +
+O Λ [( ΔA ) 4 , ( ΔB ) 4 ] ,

2

ΔΛ = Λ − Λ 0 =: l, ΔΦ = Φ − Φ 0 =: b,

ΔA = A − Λ 0 =: α, ΔB = B − B 0 =: β. (3.135)

Definition of partial derivatives:
l 10 := ∂Λ

∂A ( A 0 ,B 0 ) ,l 01 := ∂Λ ∂B ( A 0 ,B 0 ) ,

∂ 2 Λ
∂A 2 ( A 0 ,B 0 ) ,l 11 :=

∂ 2 Λ ∂ 2 Λ

∂B 2 ( A 0 ,B 0 ) ,

l 20 := 1
2 ∂A∂B ( A 0 ,B 0 ) ,l 02 := 1 2

∂ 3 Λ ∂ 3 Λ

∂B 3 ( A 0 ,B 0 ) ,

l 30 := 1
6 ∂A 3 ( A 0 ,B 0 ) ,l 03 := 1

(3.136)

6

∂ 3 Λ ∂ 3 Λ

∂A∂B 2 ( A 0 ,B 0 ) .

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

Taylor series, powers of Taylor series:
l = l 10 α + l 01 β + l 20 α 2 + l 11 αβ + l 02 β 2 + l 30 α 3 + l 21 α 2 β + l 12 αβ 2
+ l 03 β 3 +O 1 ( α 4 ,β 4 ) ,