Theory of Shape - Geometric Morphometrics for Biologists - page 92

FIGURE 4.14 Centered triangles from Figure 4.13 , scaled to
unit centroid size. Computation of centroid size is given in
Equations 4.19

3

3

4.21 . Computation of landmark coordinates after
scaling is given by Equations 4.22 and 4.23 .

2

X

1 2

W

1

These centered and scaled triangles are shown in Figure 4.14 .
Because size differences do not contribute to the differences between X pre-shape and

W pre-shape , another degree of freedom has been lost (this is the third degree of freedom
lost). In other words, size is no longer a dimension of possible variation; configurations
that differ only in size are considered equivalent. After subtracting the three degrees of
freedom representing differences in location and centroid size, we are left with three
degrees of freedom to describe differences among triangle pre-shapes

triangles that
are centered and scaled to unit centroid size. Accordingly, the pre-shape space of
triangles is a three-dimensional space. As explained above, it is the three-dimensional
surface of a four-dimensional hypersphere, so it is not an easy space to visualize or
illustrate.
To make the transition from pre-shape space to shape space, we begin by choosing one
shape and placing it in a convenient orientation; this configuration will be the reference.
For this demonstration it is convenient to use X in the orientation shown in the last few

figures. Choosing X as the reference means that W will be the target, so the next step is to

rotate ). The rotation

places it in the orientation that minimizes the difference between the two sets of landmark
coordinates ( Figure 4.15 ). After the rotation, the X - and Y -coordinates of each landmark
will be mapped to the new coordinates ( X cos

W , in the plane of the page around its centroid through some angle ( θ

θ 2 Y sin θ ), ( X sin θ 1 Y cos θ ). Thus, the

rotated form of W pre-shape will be:

2
4

3
5 (4.24)

ð 2 0 362 cos θÞ 2 ð 2 0 488 sin θÞ ð 2 0 362 sin θÞ 1 ð 2 0 488 cos θÞ

: : : :

W ð 0 516 cos θÞ 2 ð 2 0 089 sin θÞ ð 0 516 sin θÞ 1 ð 2 0 089 cos θÞ

pre - shape ; rotated 5 : : : :

ð 2 0 154 cos θÞ 2 ð 0 577 sin θÞ ð 2 0 154 sin θÞ 1 ð 0 577 cos θÞ

: : : :

Before we can pick the value of θ that will minimize the difference between the refer-

ence ( X pre-shape ) and the rotated target ( W pre-shape,rotated ), we need a criterion to define