Bridge geometry - Computational Analysis and Design of Bridge Structures - page 570

′ ′ ′ λ

( , x y z , ) = ( x y z , , ) (18.11)

where λ is called direction cosine matrix of local coordinate system and is
defined as

′ ′ ′

x x x

x y z

′ ′ ′

λ = y y y (18.12)

x y z

′ ′ ′

z z z

x y z

18.6.7.3 Transformation between two coordinate systems
Once a local coordinate system is defined, λ is known. A point represented
by a local coordinate system (

′ ′ ′ can be transformed to the global sys-
tem by Equation 18.11 so its coordinates in the global system ( ,

x y z , , )

x y z can
be obtained. Similar to Equation 18.11, its reverse transformation can be
found as

, )

( x y z

′

,

′

,

′

) ( , x y z , )

λ 1

= (18.13)

By using Equation 18.13, a point represented by its global coordinate sys-
tem ( , x y z can be transformed to a local system so its local coordinate

system (

, )

′ ′ ′ can be obtained.
Transforming Equation 18.11 to 18.13 is based on that the two coordi-
nate systems have the same origin as shown in Figure 18.22. When apply-
ing to transformation between the global and local systems as shown in
Figure 18.19, the origin of the local coordinate system has to be translated
to the same origin with the global system before the transformation. The trans-
formed coordinates will then be translated back to the true origin of the
local system.

x y z , , )

18.6.7.4 Definition of the casting system in global system
The connecting nodes between segments on the theoretical casting curve are
known, as shown in Figure 18.16. Therefore, the origin and the longitudinal
axis of the local system (Long′) for the current casting segment as shown in
Figure 18.20 can be established. As the vertical axis of the local system (Alt′)
cannot be generally assumed being parallel to the global vertical axis (Alt) due
to the existence of superelevation, the transverse axis of the local system
(Lat′) has to be defined instead.
Because the transverse axis (Lat′) is always parallel to the bulkhead, Lat′
can be known as long as a point along the bulkhead is known. This can be
obtained by the control point shown in positive Lat′ axis in Figure 18.20.
By constructing a line on the plane constructed by points p p

, − 1 and the n