Biology Reference
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We can infer from the coordinates that the two triangles have different locations, as sug-
gested in the figure. We confirm this by calculating the coordinates of the centroid using
Equation 4.5 , reproduced here:
K X
K
1
X C
X j
5
j
1
5
(4.15)
K X
K
1
Y C
5
Y j
j
1
5
For triangle
X
, the coordinates of the centroid are X C 5
(1/3)(
1
1
0)
0, and
2
1
1
5
Y C 5
(1/3)(
1
1
1)
0 . 333 . For triangle
W
, the coordinates of the centroid are
2
12
1
52
X C 5
0 . 513.
We use the coordinates of the centroid to form the centered configuration matrix
(1/3)(1 . 07
3 . 10
1 . 55)
1 . 907
and
Y C 5
(1/3)(
1 . 64
0 . 72
0 . 82)
1
1
5
2
12
1
52
XC
by subtracting the centroid coordinate from the corresponding coordinate of each
landmark:
2
3
ð
X 1 2
X C Þð
Y 1 2
Y C Þ
4
5
ð
X 2 2
X C Þð
Y 2 2
Y C Þ
XC 5
(4.16)
^
^
ð
X K
X C
Þð
Y K
Y C
Þ
2
2
This produces the centered configuration matrices:
2
3
2
3
ð 2
1
0
Þ 2
1
2 ð 2
0
333
ÞÞ
1
0
667
2
:
2
2
:
4
5 5
4
5
X centered 5
ð
1
0
Þ 2
1
2 ð 2
0
333
ÞÞ
1
667
01
0
(4.17)
2
:
2
:
ð
0
0
Þ
ð
1
2 ð 2
0
333
ÞÞ
333
2
:
:
and
2
4
3
5 5
2
4
3
5
ð
1
07
1
907
Þ 2
1
64
2 ð 2
0
513
ÞÞ
0
837
1
127
:
2
:
:
:
2
:
2
:
W
ð
3
10
1
907
Þ 2
0
72
2 ð 2
0
513
ÞÞ
1
193
0
207
(4.18)
5
:
2
:
:
:
:
2
:
centered
ð
1
55
1
907
Þ
ð
0
82
2 ð 2
0
513
ÞÞ
0
357
1
333
:
2
:
:
:
2
:
:
The centered triangles are shown in Figure 4.13 . One consequence of centering is that
the two triangles are now superimposed ; another is the loss of two degrees of freedom.
Knowing that the centroid has coordinates (0, 0), which are the means of the landmark
coordinates, we can use the coordinates of any two landmarks to determine the coordi-
nates of the third landmark. Accordingly, the space of centered triangles (which we have
not discussed previously) is a four-dimensional space. Another way to think of this is that
the two coordinates of the centroid, specifying the location of the triangle, account for two
of the six dimensions of the configuration space. Also, now that all individuals have the
same value for their centroid coordinates, the variation due to position disappears, collaps-
ing that dimension of variation to a point at the origin.
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