Biology Reference
In-Depth Information
space. As discussed in a later section of this chapter, it is possible to map locations in
Kendall's shape space to locations in a Euclidean space tangent to Kendall's shape space.
Like planar maps of the Earth, the Euclidean “maps” of shape space distort the relative
positions of shapes far from the tangent point. This becomes important when comparing
extremely dissimilar shapes. In most biological studies, the range of shapes will be small
relative to the curvature of the space, so the distortion will be mathematically trivial for
any well-considered choice of the tangent point (we discuss criteria for selecting the tan-
gent point in a later section). If you are comparing such highly dissimilar shapes that you
need to work in Kendall's shape space, you will need a more detailed understanding of
this space than presented here. The excellent texts by
Dryden and Mardia (1998)
and
Small (1996)
discuss the variables and procedures for carrying out inference in Kendall's
shape space.
Finding the Angle of Rotation That Minimizes the Euclidean Distance
Between Two Shapes
To determine the angle of rotation required to place one pre-shape at a minimum
Procrustes distance from a second, it is sufficient to rotate the first shape (the target) to
minimize the summed squared distance between it and the reference. This distance we are
minimizing is the partial Procrustes distance. Because the Procrustes distance is a mono-
tonic function of the partial Procrustes distance, this minimization of the partial Procrustes
distance also minimizes the Procrustes distance.
An arbitrary rotation of the target form (of two-dimensional landmarks,
M
2) by an
5
angle
θ
maps the paired landmarks (
X
Tj
,
Y
Tj
) of the target to the coordinates ((
X
Tj
cos
)). The sum of the squared Euclidean distances
between the
K
landmarks of this rotated target and the reference is:
θ
2
Y
Tj
sin
θ
), (
X
Tj
sin
θ
1
Y
Tj
cos
θ
X
k
D
2
2
2
5
1
½ð
X
Rj
2
ð
X
Tj
cos
θ
2
Y
Tj
sin
θÞÞ
1
ð
Y
Rj
2
ð
X
Tj
sin
θ
1
Y
Tj
cos
θÞÞ
(4.11)
j
5
where (
X
Rj
,
Y
Rj
) are the coordinates of the landmark in the reference. To minimize this
squared distance as a function of
θ
θ
, we take the derivative with respect to
and set it
equal to zero:
X
K
2
ð
X
Rj
2
ð
X
Tj
cos
θ
2
Y
Tj
sin
θÞÞð
2
X
Tj
sin
θ
2
Y
Tj
cos
θÞ
0
(4.12)
2
5
1
2
ð
Y
Rj
2
ð
X
Tj
sin
θ
1
Y
Tj
cos
θÞÞð
X
Tj
cos
θ
2
Y
Tj
sin
θÞ
j
5
1
and solve for
θ
:
arctangent
P
j
5
1
Y
Rj
X
Tj
2
!
X
Rj
Y
Tj
θ
5
(4.13)
P
j
5
1
X
Rj
X
Tj
1
Y
Rj
Y
Tj
which gives us the angle by which to rotate the target to minimize its distance from the
reference.
