Biology Reference
In-Depth Information
(1/3)(
2
1
1
1
1
0)
5
0, and the
Y
-coordinate of the centroid is (1/3)(
2
1
12
1
1
1)
52
0.333.
For the other triangle, the
X
-coordinate of the centroid is (1/3)(1.07
1
3.10
1
1.55)
5
1.907 and
the
Y
-coordinate of the centroid is (1/3)(
0.513. We now subtract these
centroid coordinates from the coordinates of the landmarks. For the first triangle, this gives us:
2
1.64
12
0.72
1
0.82)
52
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
ð
1
0
Þ
2
1
2
ð
2
0
333
ÞÞ
1
667
01
0
(3.3)
5
2
:
2
:
ð
0
0
Þ
ð
1
2
ð
2
0
333
ÞÞ
333
2
:
:
And for the other this gives us:
2
4
3
5
5
2
4
3
5
ð
Þ
2
2
ð
2
ÞÞ
1
:
07
2
1
:
907
1
:
64
0
:
513
2
0
:
837
2
1
:
127
ð
3
10
1
907
Þ
2
0
72
2
ð
2
0
513
ÞÞ
1
193
0
207
(3.4)
:
2
:
:
:
:
2
:
ð
1
55
1
907
Þ
ð
0
82
2
ð
2
0
513
ÞÞ
0
357
1
333
:
2
:
:
:
2
:
:
So the two triangles are now both centered on the same coordinates (see
Figure 3.9B
).
We now need to scale them by dividing each coordinate by centroid size. The calculation
of centroid size is simple now that the centroids are at 0,0
it is the square root of the sum
of the squared coordinates. For the first triangle, centroid size is calculated as:
ð
2
q
2
2
2
2
2
2
1
0
Þ
1
ð
2
0
667
Þ
1
ð
1
0
Þ
1
ð
2
0
667
Þ
1
ð
0
Þ
1
ð
1
333
Þ
:
:
:
:
:
(3.5)
2
160
5
:
And for the second triangle it is calculated as:
ð
2
q
2
2
2
2
2
2
0
837
Þ
1
ð
1
127
Þ
1
ð
1
193
Þ
1
ð
2
0
207
Þ
1
ð
0
357
Þ
1
ð
1
333
Þ
:
:
:
:
:
:
(3.6)
2
311
5
:
So we now divide each coordinate by centroid size. For the first triangle, dividing each
coordinate by 2.16 gives us:
2
4
3
5
5
2
4
3
5
2
1
2
0
:
667
2
0
:
463
2
0
:
309
1
1
2
0
:
667
0
:
463
2
0
:
309
(3.7)
2
160
:
0
1
333
0
000
0
617
:
:
:
And for the second triangle, dividing each coordinate by 2.311 gives us:
2
4
3
5
5
2
4
3
5
2
0
:
837
2
1
:
127
2
0
:
362
2
0
:
488
1
5
1
:
193
2
0
:
207
0
:
516
2
0
:
089
(3.8)
2
311
:
0
357
1
333
0
154
0
577
2
:
:
2
:
:
So now we have centered and scaled the triangles (see
Figure 3.9C
).
The next step is to rotate the triangles to minimize the sum of the squared differences
of the coordinates (summed over all three coordinates). To do this, we will pick one of
the triangles to serve as the reference; we will arbitrarily pick the first one. So we now
