Biology Reference
In-Depth Information
The magnitude of this vector is:
s
X
2
X
j
1α
2
Y
j
Þ
ðγ
(5A.21)
j
Using the definitions of
α
and
γ
to rearrange this and simplify it, we get:
s
γ
2
X
j
2
X
j
p
γ
p
αγðα1γÞ
p
αγ
5
X
j
1α
Y
j
5
2
α1α
2
γ
5
5
(5A.22)
So if we normalize
V
2
, we get:
K
α
p
52
γ
V
2
α
αγ
V
0
2
5
X
j
;
Y
j
(5A.23)
p
p
αγ
j
5
1
r
X
j
;
γ
α
K
r
Y
j
α
γ
52
(5A.24)
j
5
1
which is now a unit vector describing a compression/dilation operation followed by
Procrustes superimposition.
Similarly, we start with a shearing operation,
S
1
(
λ
), and corresponding Procrustes
superimposition,
P
Z
0
, to find the unit vector corresponding to these operations. First we
need to find
P
Z
0
for the
S
1
(
λ
) mapping:
X
ZZ
0
5
ð
X
j
1
iY
j
Þ 3 ð
X
j
1λ
2
iY
i
Þ
Y
j
(5A.25)
j
X
ð
X
j
1
Y
j
1
iY
i
λÞ
5
X
j
Y
j
λ1
(5A.26)
j
As before,
X
j
X
0 and
X
j
ð
X
j
1
Y
j
Þ 5
Y
j
5γ
;
1
;
X
j
Y
j
5
j
ZZ
0
5
1
1
i
γλ
(5A.27)
Also:
X
j
ð
Z
0
Z
0
5
X
j
1λ
Y
j
1
iY
j
Þ 3 ð
X
j
1λ
Y
j
2
iY
j
Þ
(5A.28)
X
X
2
Y
j
5
ð
X
j
1
2
Y
2
Y
j
Þ 5
ð
X
j
1λ
Y
j
Þ
1
2
λ
X
j
Y
j
1λ
1
1
(5A.29)
j
j
Therefore:
ZZ
0
Z
0
Z
0
5
ZZ
0
1
Pz
0
5
5
1
1
i
γλ
(5A.30)
