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ʵ
X
,
Z
0
). These determine
the position of the ellipsoidal center and are thereby called the positioning param-
eters of the reference ellipsoid.
Based on the different approaches for obtaining
ʵ
Y
,
ʵ
Z
, and so on,
ʾ
0
,
ʷ
0
, N
0
can be defined by
Δ
X
0
,
Δ
Y
0
,
Δ
ʷ
0
, N
0
, we can classify the
orientation methods into single astronomical position datum orientation and
astronomical-geodetic orientation. When a datum is oriented by a single astronom-
ical point, we simply take
ʾ
0
,
0.
The above equations have shown that, at the geodetic origin, the direction of the
normal to the ellipsoid coincides with that of the plumb line and the ellipsoid is
tangent to the geoid. It follows from (
7.26
) that:
ʾ
0
ﾼ
0,
ʷ
0
ﾼ
0, N
0
ﾼ
L
0
ﾼ ʻ
0
, B
0
ﾼ ˆ
0
, A
0
ﾼ α
0
, H
0
ﾼ
H
orthometric0
:
It can be seen that the single astronomical position datum orientation, in nature,
is to consider the astronomical longitude, latitude, and azimuth measured at the
geodetic origin as the geodetic longitude, latitude, and geodetic azimuth. The
orthometric height (or normal height) of the geodetic origin is considered the
geodetic height. Using this method it is difficult to make the ellipsoid fit the
geoid within a large area (see, e.g., DMA 1984). Hence, after basically completing
the national astro-geodetic survey, we tend to reorient by making use of the
observational
N
2
results on the condition that
ﾼ
minimum. This
is
the
∑
astronomical-geodetic orientation.
The ellipsoid can be positioned and oriented by the method of astronomical-
geodetic orientation, which provides arc measurement equations at multiple astro-
geodetic points and obtains
ʾ
0
,
ʷ
0
, and N
0
by adjustment computations.
7.3.2 Arc Measurement Equation
Arc measurements can be categorized as ancient, neoteric, and modern.
In ancient societies, when people started realizing that the Earth is a sphere, they
could then technically estimate the shape and size of the Earth by the length of an
arc between two points and the measurement of difference in latitude at the two
points. This was arc measurement in early times.
The first documented measurement of the size of the Earth was by the Hellenic
scholar Eratosthenes (276-194 BC). He estimated that the radius of the Earth was
6,844 km. Since no field observations were carried out, this could not be considered
an actual arc measurement. The first country to carry out an actual arc measurement
was China. In 724 AD (Kaiyuan 12 years, Tang Dynasty), presided over by the
Chinese astronomer Yixing (birth name Zhang Sui), the imperial astronomer Nan
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