Biomedical Engineering Reference
In-Depth Information
Fig. 11.1
The intersection
of two planes is a straight line
Fig. 11.2
An illustration
of sets of parallel lines
deforming into sets of parallel
lines and a parallelogram
deforming into a
parallelogram
2
3
3
p
10
020
001
4
5
:
F
¼
Solution
: A sketch of the set of parallel lines given by 2
X
I
þ
3
X
II
¼
0 and 2
X
I
þ
3
X
II
¼
5 is shown in Fig.
11.3a
. The inverse of the homogeneous deformation
F
is
given by
2
3
3
1
2
1
0
p
p
4
5
;
F
1
¼
1
0
2
0
001
L
1
3, 0]
T
. The set of deformed parallel lines
determined by the intersection of the planes,
a*
thus
a*
¼
a
¼
[2/
√
3, (3/2)
1/
√
3, 0]
T
¼
[2/
√
3, (3/2)
1/
√
with
c
¼
0 and 5, and
x
3
¼
0. This set of parallel
lines are given by 2
x
1
/
√
3
þ
((3/2)
1/
√
3)
x
2
¼
0 and 2
x
1
/
√
3
þ
((3/2)
1/
√
3)
x
2
¼
5 and are sketched
in Fig.
11.3b
.
The effect of homogeneous deformations on ellipsoids is similar. Recall that the
Cartesian equation for an ellipsoid is
X
I
X
II
X
III
a
2
þ
b
2
þ
c
2
¼
1
;
(11.5)
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