TRANSFORMATIONS BETWEEN COORDINATE SYSTEMS
73
4.3.2
The
Transformation
Matrix
In general, any linear coordinate transformation of a vector can be written, using
subscript notation, as
where
[a]
is called the transformation matrix. In the discussion that follows, two
simple approaches for determining the elements
of
[a]
are presented. The first assumes
two systems with known basis vectors. The second assumes knowledge of only the
coordinate equations relating the two systems. The choice between methods is simply
convenience. While Cartesian coordinate systems are assumed in the derivations,
these methods easily generalize to all coordinate transformations.
Determining
[a]
from the
Basis
Vectors
If the basis vectors
of
both coordinate
systems are known, it is quite simple to determine the components of
[a].
Consider
a vector expressed with components in two different Cartesian systems,
Substitute the expression for
V:
given in Equation 4.25 into Equation 4.26 to obtain
vk6k
=
aijvjs,!.
(4.27)
=
km,
one
of
the unprimed basis This must be true
for
any
v.
In particular, let
vectors (in other words,
Vk+-m
=
0
and
Vk=m
=
l),
to obtain
&,
=
a.
rm
&!.
1
(4.28)
Dot multiplication of
2:
on both sides yields
Unm
=
(6;
*
C,).
(4.29)
Notice the elements
of
[a]
are just the directional cosines between all the pairs
of
basis vectors in the primed and unprimed systems.
Determining
[a]
from the Coordinate
Eqclations
If the basis vectors are not
explicitly known, the coordinate equations relating the two systems provide the
quickest method for determining the transformation matrix. Begin by considering the
expressions for the displacement vector in the two systems. Because both the primed
and unprimed systems are Cartesian,
dF
=
dX.6.
=
d
x,e,,
'A'
(4.30)
where the
dxl
and
dxi
are the total differentials
of
the coordinates. Because Equation
4.25 holds for the components
of
any vector, including the displacement vector,
dx:
=
aijdx,.
(4.3
1)
