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Fig. 4.11 Helmert's
projection (P-P
0
) and
Pitzzetti's projection
(P-P
0
-P
0
0
)
60
00
, H
slight. Suppose
1,000 m (in most cases, the values are much smaller
than the values given here), then the difference between H and H
O
+ N is only
0.1 mm. The distance of P
0
P
0
0
is merely 30 cm, whose effect on the geodetic
longitude and latitude is only 0.01
00
, far less than the 0.3
00
error of the astronomical
measurement (the astronomical longitude and latitude corresponds to points on the
geoid). Therefore, difference between these two projection methods can be ignored
in practical cases.
The geodetic height of a point on the Earth's surface can be obtained directly
using GPS measurement, and the relationship between the surface point and its
corresponding point projected onto the ellipsoid is established by the Helmert's
projection. However, in classical geodetic survey, the ellipsoidal height is not
directly measured, but calculated using orthometric height (or normal height)
with corrections applied. Therefore, using Pizzett's projection to determine the
corresponding relationship between a surface point and its projection point on the
ellipsoid surface is theoretically rigorous. However, in practice, since the difference
between Helmert's and Pizzett's projections can be ignored and Helmert's projec-
tion avoids double projection, first to the geoid and then to the ellipsoid, which is
more convenient in application, Helmert's projection is adopted in classical geo-
detic computations.
So, as shown in Fig.
4.11
, the geodetic height can be calculated according to:
ʼ¼
¼
H
¼
H
O
þ
N,
ð4:41Þ
where H
O
is the orthometric height and N is the distance from the geoid to the
reference ellipsoid, called the geoid height (geoid undulation or geoid separation).
China adopts the normal height system and the geodetic height can be calculated
according to:
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