Geoscience Reference
In-Depth Information
triangles, and eventually to transform the spherical elements computed back into
the ellipsoidal elements.
Clearly, the key issue is to establish the corresponding relationship between the
ellipsoidal elements and the spherical elements. From Fig.
5.40
, the corresponding
ellipsoidal
and
spherical
elements
should
include
the
six
elements
B
1
, B
2
, A
1
, A
2
, l, S.
Reduced Latitude
When the elements on the ellipsoidal surface are converted into the corresponding
elements on the spherical surface, we may obtain different formulae for solutions of
the geodetic problem over long distances due to different options for the projection
conditions or different methods of integration. This section will discuss the repre-
sentative formula, i.e., the formula for the solution of the geodetic problem over
long distances put forward by Bessel in 1825 (see, e.g., Krakiwsky and Thomson
1974). Our discussion first considers reduced latitude and the transformation
between the reduced latitude and the geodetic latitude.
As shown in Fig.
5.41
, NPSN represents a meridian ellipse. Create an auxiliary
circle with its center at the center of the ellipse O, and the equatorial radius a as its
radius. Extend the ordinate line P
0
P of point P to intersect the circle at P". Join P
00
O;
then
P
00
OP
0
is known as the reduced latitude of point P, denoted by u.
Apparently, for any point P on the meridian ellipse there will be a reduced
latitude u corresponding to it. The relations of transformation between the reduced
latitude and geodetic latitude are derived as follows:
We establish a right-angled plane coordinate system XOY in Fig.
5.41
, and the
coordinates of P are given by:
∠
9
=
a
W
cos B
X
ﾼ
:
ð5:72Þ
p
1
sin B
a
W
1
b
W
;
e
2
e
2
Y
ﾼ
ﾼ
sin B
It is known from Fig.
5.41
that X
ﾼ
acosu, substituting X into the elliptical
equation
x
2
y
2
b
2
ﾼ
a
2
þ
1 results in Y, hence:
X
ﾼ
a cos u
:
ð
5
:
73
Þ
Y
ﾼ
b sin u
Comparing (
5.72
) and (
5.73
) yields:
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