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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