Geoscience Reference
In-Depth Information
The practical difference between Pizzetti's and Helmert's projection is
small. The ellipsoidal height
h
is equal to
H
+
N
within a fraction of a
millimeter. The ellipsoidal coordinates
ϕ
and
λ
, with respect to the two
projections, are related by the equations
ϕ
Helmert
=
ϕ
Pizzetti
+
H
R
ξ,
λ
Helmert
=
λ
Pizzetti
+
H
(5-18)
R
η
sec
ϕ,
which can be read from Fig. 5.4, since
QQ
0
=
Hε
;
R
= 6371 km is the
mean radius of the earth. Even if
ε
=1arcminuteand
H
= 1000 m, the
distance
QQ
0
is only about 30 cm and the ellipsoidal coordinates differ by
less than 0
.
01
, which is below the accuracy of astronomical observations.
For most purposes, we may, therefore, neglect the difference between the two
projections.
Pizzetti's projection is better adapted to the geoid, because there is an
exact correspondence between a geoidal point
P
0
and an ellipsoidal point
Q
0
. Helmert's projection has overwhelming practical advantages, notably
the straightforward conversion of the ellipsoidal coordinates
ϕ, λ, h
into rect-
angular coordinates
x, y, z
; it is also simpler in other respects. The decisive
advantage of Helmert's projection is its direct relation to GPS. It is, there-
fore, exclusively used now in practice.
5.6
Coordinate transformations
5.6.1
Ellipsoidal and rectangular coordinates
We now derive the relation between the ellipsoidal coordinates
ϕ, λ, h
and
the corresponding rectangular coordinates
x, y, z
.
The equation of the reference ellipsoid in rectangular coordinates is
x
2
+
y
2
a
2
+
z
2
b
2
=1
.
(5-19)
The representation of this ellipsoid in terms of ellipsoidal coordinates is given
by
x
=
N
cos
ϕ
cos
λ,
y
=
N
cos
ϕ
sin
λ,
(5-20)
z
=
b
2
a
2
N
sin
ϕ,
