Geoscience Reference
In-Depth Information
that is
X
=
x
+
h
cos
ϕ
cos
λ,
Y
=
y
+
h
cos
ϕ
sin
λ,
(5-26)
Z
=
z
+
h
sin
ϕ.
By (5-20), this becomes
X
=(
N
+
h
)cos
ϕ
cos
λ,
Y
=(
N
+
h
)cos
ϕ
sin
λ,
Z
=
b
2
(5-27)
a
2
N
+
h
sin
ϕ.
These equations are the basic transformation formulas between the ellip-
soidal coordinates
ϕ, λ, h
and the rectangular coordinates
X, Y, Z
of a point
outside the ellipsoid. The origin of the rectangular coordinate system is the
center of the ellipsoid, and the
z
-axis is its axis of rotation; the
x
-axis has
the Greenwich longitude 0
◦
and the
y
-axis has the longitude 90
◦
east of
Greenwich (i.e.,
λ
=+90
◦
).
A possible source of confusion is that the normal radius of curvature of
the ellipsoid and the geoidal undulation are both denoted by the symbol
N
;
in (5-27),
N
is, of course, the normal radius of curvature. Generally, let the
context decide between quantities of such different magnitude (6000 km and
60 m).
Equations (5-27) permit the computation of rectangular coordinates
X, Y, Z
from the ellipsoidal coordinates
ϕ, λ, h
.
The inverse procedure, the computation of
ϕ, λ, h
from given
X, Y, Z
,is
frequently performed iteratively, although a solution in closed form exists.
A possible iterative procedure is as follows.
Denoting
√
X
2
+
Y
2
by
p
, we get from the first two equations of (5-27)
or from Fig. 5.5
p
=
X
2
+
Y
2
=(
N
+
h
)cos
ϕ,
(5-28)
so that
p
cos
ϕ
− N.
h
=
(5-29)
The third equation of (5-27) may be transformed into
Z
=
N −
N
+
h
sin
ϕ
=(
N
+
h − e
2
N
)sin
ϕ,
a
2
b
2
−
(5-30)
a
2
