Biology Reference
In-Depth Information
FIGURE 4.16
Triangles
X
and
W
after rotation of
W
to minimize
the Procrustes distance. Computation of the landmark coordinates of
W
3
after rotation is given in
Equation 4.24
; the result is given in
Equation 4.26
. Vertices are numbered to indicate their homology.
W
X
2
1
The distance minimized above is the partial Procrustes distance, so we will label it
D
p
from this point forward. The value of
D
p
in this particular case is:
2
2
2
2
D
p
5
½ð
2
0
502
2
ð
2
0
463
ÞÞ
1
ð
2
0
341
2
ð
2
0
309
ÞÞ
1
ð
0
458
0
463
Þ
1
ð
2
0
254
2
ð
2
0
309
ÞÞ
:
:
:
:
:
2
:
:
:
1
1
ð
0
:
044
2
0
Þ
1
ð
0
:
596
2
0
:
617
Þ
2
5
0
:
089
(4.28)
This is the minimum length of the chord connecting the pre-shape fibers of
X
and
W
in
the pre-shape space of triangles. Because
W
is superimposed to meet the criterion of mini-
mizing the partial Procrustes distance,
W
pre-shape
,
rotated
is said to be in
partial Procrustes
superimposition
on the reference form
X
pre-shape
. We can solve for the Procrustes distance,
the arc length across the surface between
W
pre-shape,rotated
, because the radius
of the hypersphere is constrained to be one. The perpendicular from the chord to the cen-
ter of the hypersphere bisects the angle
X
pre-shape
and
(
Figure 4.17
), which has the same value (in
radians) as the arc length. Thus, there is a very simple relationship between
D
p
and
ρ
ρ
; spe-
cifically,
are so small they cannot be distin-
guished with fewer than 4 decimal places (0.08941 and 0.08943, respectively), which is not
surprising given that
ρ
5
2 arcsin (
D
p
/2). In our example,
D
p
and
ρ
represents a very small angle of just 5.1
.
Because rotational effects do not contribute to the differences between
ρ
X
pre-shape
and
W
pre-shape, rotated
, another degree of freedom has been lost (the fourth). Rotation, or ori-
entation, is no longer a dimension of possible variation; configurations that differ only
by rotation are considered equivalent. After subtracting the four degrees of freedom
representing differences in location and centroid size and rotation, we are left with
two degrees of freedom to describe differences among triangles. Accordingly, the
shape space of triangles is a two-dimensional space. As explained above, it is the two-
dimensional surface of a three-dimensional sphere, and is a relatively easy space to
visualize or illustrate.
X
pre-shape
and
W
pre-shape, rotated
are configurations in a shape space, but they are not yet in
Kendall's shape space. To make this final transition, we need to solve for the centroid size
