Biology Reference
In-Depth Information
Now we can simplify:
Pz
0
Z
0
2
ð
1
1
i
γλÞð
X
j
1λ
Y
j
1
iY
j
Þ 2 ð
X
j
1
iY
j
Þ
Z
V
1
5
5
(5A.31)
λ
λ
2
Y
j
X
j
1λ
Y
j
1
iY
j
1
i
γλ
X
j
1
i
γλ
1
i
2
γλ
Y
j
2
X
j
2
iY
j
5
(5A.32)
λ
5
λ
Y
j
1
i
γλ
X
j
2γλ
Y
j
5
Y
j
1
i
γ
X
j
2γ
Y
j
(5A.33)
λ
This leads to the series of coordinate pairs:
5 ð
Y
j
ð
1
2γÞ; γ
X
j
Þ
(5A.34)
or
V
1
5 ðα
Y
j
; γ
X
j
Þ
(5A.35)
The magnitude of this vector is:
s
s
α
X
j
ðα
2
X
j
2
X
j
p
α
2
Y
j
1γ
2
X
j
Þ
5
Y
j
1γ
X
j
5
2
γ 1γ
2
α
(5A.36)
p
αγðα1γÞ
p
αγ
5
5
(5A.37)
V
1
obtained by normalizing V
1
is:
so the unit vector
r
X
j
γ
α
K
r
Y
j
;
α
γ
K
p ;
γ
α
Y
1
α
X
j
αγ
V
0
1
5
1
5
(5A.38)
p
j
5
j
5
1
which may now be used to determine the shear component of the uniform deformation.
Some software packages will give you
as used in the calculation of the uniform
component, others may give you the unit vectors instead. The expressions are for coordi-
nates of the unit vectors for shear and compression/dilation for a reference form rotated
to principal axis orientation. It turns out to be straightforward to rotate them to unit
vectors to match any reference orientation preferred by a researcher, although some pro-
grams may not offer this option, meaning that the reference may be oddly oriented by the
software.
α
and
γ
Calculating Uniform Components Based on Other Superimpositions
The approach taken in the above derivation was to determine the unit vectors that
would result from a shear or compression/dilation of a reference form, followed by
Procrustes superimposition back onto the reference form. It is also possible to determine
the unit vectors produced by a shear or compression/dilation of a reference, followed by
sliding baseline registration (SBR) or a two-point registration that yields Bookstein coordi-
nates (BC). These unit vectors and specimens can then be used in SBR or BC to calculate
the uniform components of the deformation,
just as we did with those in Procrustes
