Civil Engineering Reference
In-Depth Information
Y
a
y
a
r
a
x
0
X
a
z
Z
Figure 18.22
Direction cosines.
transformation of a fixed point from one coordinate system to another is
briefly summarized.
18.6.7.1
Direction cosines
Given a direction
a
with a length of
r
and coordinate components of
a a a
x
, , as shown in Figure 18.22, the cosines of its angles to three axes
a r a r a r
x
y
z
, , , respectively, are called direction cosines. Specifically when
a
is a unit direction, it can be represented by its direction cosines as
y
z
a
=
(
a a a
x
,
,
z
)
(18.9)
y
Having a direction's cosines as shown in Equation 18.9, the point on direc-
tion
a
with a length ordinate of
l
a
can be represented by
l
(
a a a
,
,
)
.
a
x
y
z
18.6.7.2
Direction cosines matrix of a
local coordinate system
Similar to Equation 18.9, direction cosines of three axes of a defined local
system
X Y Z
′ ′ ′
,
,
can be found as
X
′
=
(
x x x
′
,
′
,
′
)
;
Y
′
=
(
y y y
′
,
′
,
′
)
;
Z
′
=
(
z z z
′
,
′
,
′
)
(18.10)
x
y
z
x
y
z
x
y
z
Given a point
P
, which has coordinates of
(
′ ′ ′
in a local coordinate
system defined as Equation 18.10, its representation in the global coordi-
nate system where the local coordinate system is defined as
x y z
,
,
)
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