Geoscience Reference
In-Depth Information
longitude
. The ellipsoidal-harmonic coordinate
β
is the
reduced
latitude, and
the spherical coordinate
ϕ
is the
geocentric
latitude.
The latitude
ϕ
refers to the
reference ellipsoid
. The reduced latitude
β
refers to the
coordinate ellipsoid u
= constant (confocal ellipsoid through
P
in Fig. 5.6).
So far so clear. Real attention is necessary when using the coordinate
ϑ
,
which has been introduced as complement of the spherical coordinate
ϕ and
as the complement of the ellipsoidal harmonic
β
as well.
Therefore, a correct but clumsy notation would be
ϑ
ellipsoidal-harmonic
=90
◦
−
β,
(5-39)
=90
◦
−
ϑ
spherical
ϕ.
Note, however, that we did not use these indications to distinguish be-
tween the spherical and the ellipsoidal-harmonic
ϑ
! Thus, the reader is chal-
lenged to attentively distinguish between these quantities. Wherever possi-
ble, we tried to avoid conflicts.
Some examples: we used the spherical coordinates
r, ϑ, λ
in Sects. 1.4,
1.11, 1.12, 1.14, 2.5, 2.6, 2.13, 2.18, etc. We used the ellipsoidal-harmonic
coordinates
u, ϑ, λ
in Sects. 1.15, 1.16; we used the ellipsoidal-harmonic co-
ordinates
u, β, λ
in Sects. 2.7, 2.8, and we used the spherical coordinates
r, ϑ, λ
as well as the ellipsoidal-harmonic coordinates
u, β, λ
in Sect. 2.9.
The following equations express the rectangular coordinates in these
three systems:
X
=(
N
+
h
)cos
ϕ
cos
λ
=
√
u
2
+
E
2
cos
β
cos
λ
=
r
cos
ϕ
cos
λ,
Y
=(
N
+
h
)cos
ϕ
sin
λ
=
√
u
2
+
E
2
cos
β
sin
λ
=
r
cos
ϕ
sin
λ,
Z
=
b
2
(5-40)
a
2
N
+
h
sin
ϕ
=
u
sin
β
=
r
sin
ϕ.
These relations, which follow from combining Eqs. (1-26), (1-151), and (5-
27), can be used if we wish to compute
u
and
β
from
h
and
ϕ
or from
r
and
ϕ
,etc.
5.7
Geodetic datum transformations
5.7.1
Introduction
First we define a
geodetic datum
or a
geodetic reference system
.Itisde-
fined by (1) the dimensions of the reference ellipsoid (semimajor axis
a
and
