Establishment of Geodetic Coordinate Systems - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 325

0
@

1
A ﾼ

0
@

1
A

0
@

1
A ,

X 0
Y 0
Z 0

cos α 1 cos ʲ 1 cos ʳ 1 X

Y
Z

cos α 2 cos ʲ 2 cos ʳ 2 ð7:2Þ

cos α 3 cos ʲ 3 cos ʳ 3

where the coefficients are termed the transformation coefficients. The first, second,
and third row of the coefficient matrix are the respective coordinates of i 0 , j 0 , andk 0
in O

XYZ, and i 0 i 0 ﾼ 1, j 0 j 0 ﾼ

k 0

k 0 ﾼ 1, i 0 j 0 ﾼ 0, i 0 k 0 ﾼ 1, 0, and

‐

k 0 ﾼ

j 0

0. Hence, the nine direction angles in ( 7.2 ) should satisfy the six relational
expressions below:

9
=

cos 2 cos 2 cos 2 α 1 þ ʲ 1 þ ʳ 1 ﾼ 1

cos 2 cos 2 cos 2 α 2 þ ʲ 2 þ ʳ 2 ﾼ 1

cos 2 cos 2 cos 2

α 3 þ ʲ 3 þ ʳ 3 ﾼ 1

: ð 7 : 3 Þ

cos α 1 cos α 2 þ cos ʲ 1 cos ʲ 2 þ cos ʳ 1 cos ʳ 2 ﾼ 0 ;

cos α 2 cos α 3 þ cos ʲ 2 cos ʲ 3 þ cos ʳ 2 cos ʳ 3 ﾼ 0

cos α 3 cos α 1 þ cos ʲ 3 cos ʲ 1 þ cos ʳ 3 cos ʳ 1 ﾼ 0

Theoretically, only three direction angles out of the nine are independent,
meaning that one can employ any three arbitrary independent direction angles to
represent the remaining six. When studying the positioning and orientation of the
ellipsoid as well as the transformation between different coordinate systems, we are
more concerned with the angles between the corresponding coordinate axes. There-
fore, we have chosen ʲ 2 (the angle between

the two Y-axes), and ʳ 3 (the angle between the two Z-axes). Of these, ʲ 2 and ʳ 3 are
our top concerns. Since the Z and Z 0 axes coincide with their respective minor axis
of the ellipsoid,

α 1 (the angle between the two X-axes),

ʳ 3 then denotes the angle between the minor axes of two ellipsoids
that are non-parallel to each other. ZOX and Z 0 OX 0 are their respective planes of
initial geodetic meridian. Hence, ʲ 2 represents the angle between two planes of

initial geodetic meridian that are not parallel to each other.

7.1.2 Coordinate Transformations in Terms of Euler Angles
as Rotation Parameters

The formulae are long-winded if ʳ 3 are chosen as the independent

direction angles. Thus, we opt for another three mutually independent parameters to
represent the direction angles. These three parameters are the angles produced by
three successive rotations about the coordinate axis, defined as the Euler angles (see
e.g., Grewal et al. 2001). Euler angles are different from the angles between the
corresponding axes of two Cartesian coordinate systems, but an analytical relation-
ship can be established between them. The Euler angles in the geodetic coordinate
system are also referred to as rotation parameters.

α 1 , ʲ 2 , and