Biology Reference
In-Depth Information
FIGURE 4.6 An alternative visualization of the concept of a fiber. Each column shows rotations of a single
shape; triangles in different columns differ in shape. Each column represents a single fiber.
represent the number of dimensions needed to describe shapes of triangles, as explained
in the next section.)
With the concept of fiber in hand, it is now possible to talk about the separation of
shapes and the distance between them. Figure 4.7 shows the same two fibers on the
curved surface of the pre-shape space hypersphere as in Figure 4.5 . In addition, Figure 4.7
shows an arc (
) crossing the surface from one fiber to the other, and the chord ( D p ) that
passes through the interior of the hypersphere between the same two surface points. We
can draw many such arcs connecting a rotation of the pre-shape
ρ
Z 1
with a rotation of the
pre-shape
that is, the one connecting fibers at
their “point of closest approach”. Finding the shortest possible distance between points is
a common tactic for defining distances between objects in spaces. When we find that dis-
tance, we will find the rotation that is optimal in the sense of being the minimum distance
between shapes. The length of this arc is known as the Procrustes distance , and it is quanti-
fied by determining the angle between the radii that connect the center of the hypersphere
to the point at which the fibers most closely approach each other. Figure 4.8 shows the
cross-section through the pre-shape space in the plane defined by those two radii. The
angle subtended by the arc is
Z 2
. The arc we want is the shortest one
ρ
(in radians) times the length of the radius. Because we have constrained the radius to a
length of one, the length of the arc is the value of the angle. This value ranges from zero to
π
ρ
; the chord length is D p . The length of the arc is equal to
/2 , the hemisphere may always be oriented so that one specimen is at the pole, the
farthest location the second may be located at is the equator.
/2 ;at
π
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