Establishment of Geodetic Coordinate Systems - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 330

Fig. 7.3 Transformations
between geodetic Cartesian
coordinate systems

2
4

3
5

Δ X 0

Δ Y 0

2
4

3
5

2
4

3
5

2
4

3
5

X
Y
Z

X
Y
Z

1000 Z old Y old X old Δ Z 0

ʵ X
ʵ Y
ʵ Z
Δ

ﾼ 010Z old 0 X old Y old :

001 Y old X old 0 Z old

new old

m

ð 7 : 14 Þ

0, then it is called the three-parameter
formula, indicating that the scales of the two geodetic Cartesian coordinate systems
are consistent and the corresponding coordinate axes are mutually parallel. Like-
wise, in ( 7.13 ), by leaving out certain parameters, we can obtain the four-parameter,
five-parameter, and six-parameter transformation formulae.
In order to obtain the seven transformation parameters in ( 7.14 ), at least three
points with two sets of coordinates, both the old and the new (known as common
points) are needed. The transformation parameters will be solved according to the
principle of adjustment.
Actually, the accuracy of the common point coordinates and other factors such
as the number and geometric distribution of the common points all exert an
influence on the solution of transformation parameters. Thus, in practical cases,
we have to choose a certain number of common points with relatively high accuracy
and with even distribution and wide coverage.
After carrying out adjustment computations of the difference between the new
and old coordinates as the observed quantity in ( 7.14 ), we will obtain the correction
to the observed quantity. This shows that the new coordinates transformed from the
old coordinates of the common points according to ( 7.14 ) are not completely

In ( 7.13 ), if ʵ X ﾼ ʵ Y ﾼ ʵ Z ﾼ 0 and Δ m ﾼ