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#
2
3
2
dy
dx
1
þ
x
0
R
x
0
ﾼ
:
d
2
y
dx
2
x
0
Derivations of the Formula
Transformation Between Coordinate Systems P-xyz and O-XYZ
The equation of the ellipsoid in the coordinate system P-xyz cannot be computed
unless the relationship of transformation between the coordinate systems O-XYZ
and P-xyz is determined. As shown in Fig.
5.10
, let P be a point on the initial
meridian (the coordinate plane XOZ); PK is the normal passing through P, PK
ﾼ
N,
e
2
), and the coordinate values of point P can be obtained from
PQ
ﾼ
N(1
Fig.
5.10
:
2
3
2
3
X
Y
Z
N cos B
0
4
5
ﾼ
4
5
:
e
2
N 1
ð
Þ
sin B
As shown in Fig.
5.10
, in order to make point P coincide with point O, the origin
of the coordinate system P should be translated to the point O. To make the
coordinate plane xPz coincide with the meridian, the coordinate system should be
rotated with a negative angle A around the z-axis. Similarly, for the z-axis, which is
directed towards the normal, parallel to the minor axis of the ellipsoid, the coordi-
nate system should be rotated by an angle of 90
+ B about the y-axis. P-xyz is
thereby transformed into O-XYZ, given by:
2
3
2
3
2
3
X
Y
Z
x
y
z
N cos B
0
R
z
4
5
ﾼ
R
y
90
4
5
þ
4
5
þ
B
ðÞ
A
e
2
N 1
ð
Þ
sin B
2
3
2
3
sin B cos A
sin B sin A
cos B
x
y
y
4
5
4
5
ﾼ
sin A
cos A
0
cos B cos A
cos B sin A
sin B
2
3
N cos B
0
4
5
:
þ
ð
5
:
24
Þ
e
2
N 1
ð
Þ
sin B
Namely:
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