Biomedical Engineering Reference
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respectively, and
R
ðy
12
Þ
and
R
ðy
23
Þ
are given by
2
4
p
sin 2
y
cos 2
y
3
5
sin
2
2
y
cos
2
2
y
0
0
0
p
sin 2
y
cos 2
y
sin
2
2
y
cos
2
2
y
0
0
0
0
0
1
0
0
0
0
0
0
cos 2
y
sin 2
y
0
0
0
0
sin 2
y
cos 2
y
0
p
sin 2
p
sin 2
sin
2
2
cos
2
2
y
cos 2
y
y
cos 2
y
0
0
0
y
y
and
2
4
3
5
1
0
0
0
0
0
p
sin 2
y
cos 2
y
sin
2
2
y
cos
2
2
y
0
0
0
p
sin 2
y
cos 2
y
sin
2
2
y
cos
2
2
y
0
0
0
p
sin 2
y
cos 2
y
p
sin 2
y
cos 2
y
sin
2
2
y
cos
2
2
y
0
0
0
0
0
0
0
cos 2
y
sin 2
y
0
0
0
0
sin 2
y
cos 2
y
(4.7)
respectively. The formulas (
4.1
)-(
4.7
) provide the 3-D and 6-D orthogonal
transformations for a plane of reflective symmetry. These mirror symmetry
transformations will be used in Sects.
4.6
and
4.7
to develop the representations
in Tables
4.3
and Tables
4.4
and
4.5
for the matrices
A
and
C
, respectively, for the
various material symmetries.
Problems
4.4.1. Verify that the transformation (
4.2
)
R
ðaÞ
¼
1
2
a
a
has the properties
(
4.1
),
R
ðaÞ
a
and
R
ðaÞ
0
for all
b
.
4.4.2. Construct the orthogonal transformations
R
ðe
3
Þ
and
a
¼
b
¼
b
where
a
b
¼
R
ðe
3
Þ
, then verify their
orthogonality.
4.4.3. Construct the orthogonal transformations
R
ðy
13
Þ
and
R
ðy
13
Þ
associated with the
vector
a
e
3
, and verify their orthogonality.
4.4.4. Show that the reflections
R
ðy
12
Þ
and
R
ðy
12
Þ
when evaluated at
¼
cos
y
e
1
þ
sin
y
y ¼
0 coincide
R
ðe
1
Þ
, and when evaluated at
with the reflections
R
ðe
1
Þ
and
y ¼ p
/2, they
coincide with the reflections
R
ðe
2
Þ
and
R
ðe
2
Þ
.
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