464 21 Datum problems
Box 21.6 (Inverse coordinate conformal transformation (datum transformation) extended by ellipsoid parameters close to the identity ).
Inverse transformation {X, Y, Z} → {L, B, H}:
L = arctan(Y/X) +
„
−
1
2
sgn Y −
1
2
sgn Y sgn X + 1
«
π ∈ {R 0 ≤ L < 2π} ,
B = arctan
»
Z
√
X 2 + Y 2
A 1 (1 − E
2 sin
2 B)
−1/2 + H(L, B)
A 1 (1 − E 2 )(1 − E 2 sin
2 B) −1/2 + H(L, B)
–
=
= arctan
»
Z
√
X 2 + Y 2
N (B) + H(L, B)
(1 − E 2 )N (B) + H(L, B)
–
∈ {R − π/2 < B < +π/2} .
(21.31)
Inverse curvilinear coordinate conformal transformation close to the identity
(datum transformation).
1st variant:
L = arctan
“
y − t y − zα + xγ − ys
”
/
“
x − t z + zβ − yγ − xs
”
,
B = B(x, y, z, t x , t y , t z , α, β, γ, s) .
(21.32)
2nd variant:
tan L =
=
“
[a 1 /
p
1 − e 2 sin
2 b + h(l, b)] cos b sin l − t y − [a 1 (1 − e
2 )/
p
1 − e 2 sin
2 b + h(l, b)] sin bα
+[a 1 /
p
1 − e 2 sin
2 b + h(l, b)] cos b sin lγ
− [a 1 /
p
1 − e 2 sin
2 b + h(l, b)] cos b sin ls
”
/
“
[a 1 /
p
1 − e 2 sin
2 b + h(l, b)] cos b cos l − t x + [a 1 (1 − e
2 )/
p
1 − e 2 sin
2 b + h(l, b)] sin bβ
−[a 1 /
p
1 − e 2 sin
2 b + h(l, b)] cos b sin lγ
− [a 1 /
p
1 − 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 ) .
(21.33)
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 ,
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 ,
(21.34)
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 )
etc.
(21.35)
and
δa := a 1 − A 1 = −δA , δe
2 := e
2 − E
2 = −δE
2 , δe = (2e)
−1 δe
2 .
(21.36)