Datum Problems - Map Projections: Cartographic Information Systems - page 521

+[ a 1 / 1 [ a 1 / 1 e 2 sin 2 b + h ( l,b )] cos b sin lγ e 2 sin 2 b

+ h ( l,b )] cos b sin ls )

− − −

/ ([ a 1 / 1 e 2 )

e 2 sin 2 b + h ( l,b )] cos b cos l

− − t x +[ a 1 (1 −

/ 1

e 2 sin 2 b + h ( l,b )] sin bβ

−

(21.33)

[ a 1 / 1 [ a 1 / 1 e 2 sin 2 b + h ( l, b )] cos b sin lγ e 2 sin 2 b

+ h ( l,b )] cos b cos ls ) ,

− − − −

L = L ( l,b,h ( l,b ) ,t x ,t y ,t z ,α,β,γ,s,a 1 ,e 2 ) ,
B = B ( l,b,h ( l,b ) ,t x ,t y ,t z ,α,β,γ,s,a 1 ,e 2 ) .

Taylor expansion:

L = l + L t x t x + L t y t y + L t z t z + L α α + L β β + L γ γ + L s s + L a δa + L e δe 2 ,

(21.34)

B = b + B t x t x + B t y t y + B t z t z + B α α + B β β + B γ γ + B s s + B a δa + b e δe 2 ,

subject to

L t x := ∂L

∂t x ( t x = t y = t z =0 ,α = β = γ =0 ,s =0 ,a 1 = A 1 ,e 2 = E 2 )

(21.35)

etc.
and

δA, δe 2 := e 2 E 2 = δE 2 ,δe =(2 e ) − 1 δe 2 . δa := a 1 − A 1 = − − − (21.36)

Box 21.7 (Inverse curvilinear conformal coordinate transformation (datum transformation)
extended by ellipsoid parameters close to identity, Jacobi matrix).

⎡ ⎤

t x
t y
t z
α
β
γ
s
δa
δe 2

⎣ ⎦

L
B

= l +B

+O 2 LB = l

+B x +O 2 LB .

(21.37)

b b