Establishment of Geodetic Coordinate Systems - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 331

equivalent to the values of the given new coordinates. In practice, however, the
coordinate values of the given points are often required to be constant and unchang-
ing. In order to settle this issue, the transformed value of the common points can be
corrected to the known values and one can deal with the transformed values of the
non-common points, e.g., calculate the correction to the transformed values of the
non-common points using the weighted average according to the formula below:

X

n

p i v i

V 0

1

ﾼ ,

X

n

p i

1

where n denotes the number of the common points. The weight of the ith common
point can be defined by the distance (S i ) between the non-common point and the
common point. We can set p i ﾼ

1

S i 2 , v i is the correction to the coordinate value of the

ith common point, namely v i ﾼ transformed value, and the coordinate

of the common point adopts the given value. This is only one method for an
interpolation of the residuals, which is not a similar transformation since the
transformed network of identical points might lose its shape.

given value

7.2.2 Transformation Between Different Geodetic
Coordinate Systems

As indicated above, the transformation formulae for different geodetic Cartesian
coordinate systems generally involve seven parameters: three translations, three
rotations, and one scaling. For the transformation between different geodetic
coordinate systems, two additional transformation parameters are needed, namely
the different Earth ellipsoid parameters corresponding to the two types of geodetic
coordinate systems. The transformation formula for different geodetic coordinate
systems is also referred to as the geodetic coordinate differential formula or
ellipsoid transformation differential formula. When inclusive of the rotation and
scale parameters, it is called the generalized differential formula for geodetic
coordinates or generalized differential formula for ellipsoid transformation.
Given that the relationship between the geodetic Cartesian coordinates and the
geodetic coordinates of a given point in space is:

2
4

3
5 ﾼ

2
4

3
5 ,

X
Y
Z

ð N þ H Þ cos B cos L

ð N þ H Þ cos B sin L

N 1

H sin B

e 2 ð Þþ