21-1 Analysis of a datum problem 457
Box 21.3(Curvilinear coordinate conformal transfor mation (datum transformation) extended by ellipsoid
parameters, ellipsoid- of- revolution E
2
A 1 ,A 2 versus E
2
a 1 ,a 2 , ellipsoidal coordinates {l, b, h} of local type versus
ellipsoidal coordinates {L, B, H} of g
lobal type ).
A 1 , a 1 ∈ R
+ semi-major axes; E :=
p
(A
2
1 − A
2
2 )/A
2
1 , e =
p
(a
2
1 − a
2
2 )/a
2
1 relative eccentricities.
Direct transformation {l, b, h} → {x, y, z}:
x =
"
a 1
p
1 − e 2 sin
2 b
+ h(l, b)
#
cos b cos l = [n(b) + h(l, b)] cos b cos l ,
y =
"
a 1
p
1 − e 2 sin
2 b
+ h(l, b)
#
cos b sin l = [n(b) + h(l, b)] cos b sin l ,
z =
"
a 1 (1 − e
2 )
p
1 − e 2 sin
2 b
+ h(l, b)
#
sin b
= [(1 − e
2 )n(b) + h(l, b)] sin b ,
(21.13)
n(b) :=
a 1
p
1 − e 2 sin
2 b
, m(b) :=
a 1 (1 − e
2 )
p
(1 − e 2 sin
2 b) 3
.
(21.14)
Direct transformation {L, B, H} → {X, Y, Z}:
X =
"
A 1
p
1 − E 2 sin
2 B
+ H(L, B)
#
cos B cos L = [N (B) + H(L, B)] cos B cos L ,
Y =
"
A 1
p
1 − E 2 sin
2 B
+ H(L, B)
#
cos B sin L = [N (B) + H(L, B)] cos B sin L ,
Z =
"
A 1 (1 − E
2 )
p
1 − E 2 sin
2 B
+ H(L, B)
#
sin B
= [(1 − E
2 )N (B) + H(L, B)] sin B ,
(21.15)
N (B) :=
A 1
p
1 − E 2 sin
2 B
, M(B) :=
A 1 (1 − E
2 )
p
(1 − E 2 sin
2 B) 3
.
(21.16)
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
p
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
p
x 2 + y 2
n(b) + h(l, b)
(1 − e 2 )n(b) + h(l, b)
#
∈ {R − π/2 < b < +π/2} .
(21.17)
h(l, b) = h(x, y, z) (E. W. Grafarend and P. Lohse (1991)).