Biology Reference
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reference. That is, the orientation is chosen to minimize the Procrustes distance between
the target and reference. The points on those fibers that are farther from the reference dif-
fer from it in both shape and rotational effects. By selecting the point of closest approach,
we reduce each fiber of pre-shapes to a single point (a shape); consequently, configura-
tions in this set differ only in shape.
The shape space we just described has fewer dimensions than the pre-shape space from
which it was derived. The number of dimensions lost in the transition is given by:
M
ð
M
1
Þ
2
(4.10)
2
where
M
is the number of landmark coordinates. For two-dimensional landmarks,
Equation 4.10
simplifies to one, which reflects the fact that a planar shape can only be
rotated about its centroid on one axis (the axis perpendicular to the plane of the shape)
and still stay in the same plane. Consequently, shape spaces of two-dimensional configura-
tions of
K
landmarks have 2
K
4 dimensions. The four lost dimensions are those describ-
2
ing differences in size (
1). For three-dimensional
landmarks,
Equation 4.10
simplifies to three, which reflects the fact that a three-dimen-
sional shape can be rotated about its centroid on three distinct orthogonal axes in the
three-dimensional coordinate space. Subtracting three from the 3
K
1), translation (
2) and rotation (
2
2
2
2
4 dimensions of the
pre-shape space (from
Equation 4.9
) yields 3
K
7 dimensions for shape spaces of three-
dimensional shapes, which simplifies to five dimensions for the shape space of tetrahedra.
The seven lost dimensions are those describing differences in size (
2
2
1), translation (
2
3)
and rotation (
3).
In the special case of triangles, the shape spaces defined above with centroid size still
equal to one, are the familiar two-dimensional surfaces of three-dimensional spheres.
Because this is a reasonably simple geometry to visualize and illustrate, we will focus on
triangles before returning to the general case. In
Figure 4.9
, we show half of a space deter-
mined by using the equilateral triangle as the reference. Because we retain the constraints
that each triangle is centered and scaled to centroid size of one, the hemisphere has a
radius of one. For convenience, the space is oriented so that the point representing the
equilateral triangle configuration is located at the pole. At the equator with maximal dif-
ference from the reference are various reflections of the reference (
Rohlf, 1999, 2000
[see
Figure 1
]). Collinear triangles (with all points along a single line) are located at a
Procrustes distance of
2
/4 from the reference. Although the shape space just described is
a useful construction, it does not satisfy the mathematician's urge to find the smallest
distances between configurations with those shapes. To illustrate this point, we consider a
slice through the polar axis of the hemisphere of triangles just described (
Figure 4.10
).
As in pre-shape space, the distance of a shape (A) from the reference is
π
. The angle and
the arc length are unchanged because the dimension eliminated in the transition from
pre-shape space to this shape space did not contribute to the measurement of the shape
difference. It should be apparent in
Figure 4.10
that the arc across the surface is not the
shortest possible distance between the two shapes. The chord passing through the interior
of the hemisphere would be shorter, but it is still not the shortest possible distance
between configurations with those shapes. We obtain that shortest possible distance, and
the relevant configurations, by changing the constraint on the centroid sizes of the two
ρ
