Biology Reference
In-Depth Information
FIGURE 4.2
Example of a triangle.
0, 1
-1, -1
1, -1
image of the body of a piranha, and 16 landmarks on a two-dimensional image of a rat
skull, both configuration matrices are in the same configuration space. Obviously, the
landmarks on these two structures don't have any kind of homology with one another, so
comparisons would be meaningless, but the measurements are in the same configuration
space. Clearly, any group of biologically similar organisms (with matched landmarks) will
occupy a relatively small part of configuration space because the locations of their corre-
sponding landmarks will be fairly similar. For example, in the 16
2 configuration space,
3
piranhas will occupy a very small part of the space
that space also contains the 16
2
3
two-dimensional coordinates of rat skulls.
The configuration space of
K
landmarks with
M
coordinates per landmark has
K
3
M
dimensions. To specify the location of any shape in that space, we must specify
K
3
M
components of a vector (or elements in a matrix).
Position or Location of a Configuration Matrix
The position of a configuration matrix is the location of the centroid of that matrix. This
centroid is the
M
-dimensional vector (two in the case of the two-dimensional landmarks
of piranhas) whose components are the averages of the
X
and
Y
coordinates of the land-
marks (in the two-dimensional case), so the centroid position is given by:
K
X
K
1
X
C
X
j
5
j
5
1
(4.5)
K
X
K
1
Y
C
5
Y
j
j
5
1
For example,
Figure 4.3
shows the centroid position of the triangle seen earlier, which is
located at (0,
0
.
333).
A configuration matrix is said to be
centered
if the average of all the coordinates is zero.
Centering is useful because it often simplifies the mathematics; it is done by translating
the configuration along the
X
- and
Y
-axes. That translation is done by adding a constant
2
