Biology Reference
In-Depth Information
FIGURE 4.4
Centroid size of the triangle in
Figure 4.2
,
calculated as the sum: (L1
2
0, 1
L2
2
L3
2
)
1/2
2.16.
1
1
5
L3
L2
L1
1,
1
1,
1
and, in this case,
C
j
stands for the location of the
j
th component of the centroid.
C
1
is the
X
-coordinate of the centroid and
C
2
is its
Y
-coordinate.
Centroid size is thus the square root of the sum of the squared distances of the land-
marks from the centroid. The distances from the centroid to each landmark of the trian-
gle are shown in
Figure 4.4
; the centroid size of this triangle is simply the square root
of the sum of the squared lengths of these lines. Centroid size is not altered by chang-
ing the position of the configuration, because this leads to all landmarks (and the
centroid) changing by a common amount. Similarly, multiplying the configuration
matrix
X
matrix
by a constant factor increases centroid size by the same factor. Two configura-
tions of landmarks that differ only in centroid size do not differ in shape (they differ
only in scale).
X
PRE-SHAPE SPACE
As we stated above, every configuration of
K
landmarks having
M
coordinates can be
thought of as a point in a space with
K
M
dimensions. (To avoid confusion, we should
make it clear that by “point” in this context we mean an individual shape, an entire con-
figuration of landmarks, not one landmark.) Some of the configurations in this space differ
only in centroid size; others differ only in location (coordinates of the centroid). We can
define a subset of configurations that do not differ in location or size by placing two
restrictions on each configuration matrix: (1) that it be centered, and (2) that centroid size
be one. These restrictions define a space called pre-shape space (
Dryden and Mardia,
1998
). In practice, we translate and scale each of the original configurations in our data so
that the new configurations meet the restrictions of pre-shape space. In doing this, we are
using two of the three operations that do not alter shape. Each of the new configurations
is a centered pre-shape.
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