Geoscience Reference
In-Depth Information
between two neighboring tangents of this projection of the plumb line is
dϕ
=
−
κ
1
dh ,
(5-125)
where the minus sign is conventional and the curvature
κ
1
is given by (2-50):
κ
1
=
1
g
∂g
∂x
.
(5-126)
The
x
-axis is horizontal and points northward. Hence, the total change of
latitude along the plumb line between a point on the ground,
P
,andits
projection onto the geoid,
P
0
,isgivenby
δϕ
=
P
P
0
P
dϕ
=
−
κ
1
dh
(5-127)
P
0
or
P
1
g
∂g
∂x
dh .
δϕ
=
−
(5-128)
P
0
Using
κ
2
of (2-51), we similarly find for the change of longitude
P
1
g
∂g
∂y
dh ,
δλ
cos
ϕ
=
−
(5-129)
P
0
where the
y
-axis is horizontal and points eastward.
Alternative formulas
There is a close relationship between the curvature reduction of astronomical
coordinates and the orthometric reduction of leveling, considered in Sect. 4.3.
The orthometric correction
d
(OC) has been defined as the quantity that
must be added to the leveling increment
dn
in order to convert it into the
orthometric height difference
dH
:
d
(OC) =
dH − dn .
(5-130)
From Fig. 5.18, we see that, for a north-south profile, the curvature reduction
and the orthometric correction are related by the simple formula
δϕ
=
∂
(OC)
∂x
.
(5-131)
Similarly, we find
δλ
cos
ϕ
=
∂
(OC)
∂y
.
(5-132)
