Biomedical Engineering Reference
In-Depth Information
Fig. 4.9
Illustration of
mirror symmetry
b
b
c
c
a
a
Congruent tetrahedra
m
b
b
c
c
a
a
Mirror-image tetrahedra
respect to the symmetry plane is said to be
achiral
, that is to say that the objects are
mirror images. If the objects are not reflections of one another, they are said to be
chiral
. The plane with respect to which two objects have mirror symmetry is called
their
plane of reflective symmetry
. A material is said to have a
plane of reflective
symmetry
or a
mirror plane
at a point in the material if the structure of the material
has mirror symmetry with respect to a plane passing through the point.
In the following three sections the concept of a plane of reflective symmetry
will now be used to classify the various types of anisotropy possible in the 3-D and
6-D symmetric linear transformations,
the
A
in the three-dimensional
linear
t
, (A39), and
C
in the six-dimensional linear transformation
transformation
r
¼
A
T
J
, (A160). Recall that the matrices
A
and
C
transform according to the rules
(A83) and (A162), respectively, when the coordinate system is changed. In order to
apply the restrictions of reflective symmetry to (A83) and (A162), it is necessary to
have a representation for a plane of reflective symmetry for the orthogonal
transformations
Q
and
Q
, respectively. To construct such representations let
a
be
a unit vector representing the normal to a plane of reflective symmetry and let
b
be any vector perpendicular to
a
, then
a
·
b
¼ C
¼
0
for all
b
. An orthogonal transforma-
tion with the properties
R
ðaÞ
R
ðaÞ
a
¼
a
;
b
¼
b
(4.1)
represents a plane of reflective, or mirror, symmetry. The transformation (
4.1
)
carries every vector parallel to the vector
a
, the normal to the plane of mirror
symmetry, into the direction
a
and it carries every vector
b
parallel to the plane
into itself. The orthogonal transformation with the property (
4.1
) is given by
R
ðaÞ
ij
R
ðaÞ
¼
1
2
a
a
; ð
¼ d
ij
2
a
i
a
j
Þ;
(4.2)
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