Civil Engineering Reference
In-Depth Information
Figure 4.18
Region of intersection of metrics is not an ellipse.
of d or the boundary of the intersected region of several metrics is not an ellipse; thus, the
direct minimisation of metrics does not, in general, give rise to another metric. To define
a metric that is a subset of all the given metrics, we have to find the largest ellipse within
the regions of intersection. Here we will explore a less general procedure by examining the
intersection of two metrics at a time.
Our problem is as follows: given two metrics M
1
and M
2
, find the metric M that is bounded
by both M
1
and M
2
. For metrics M
1
and M
2
without common eigenvectors (θ
1
≠ θ
2
), intro-
duce auxiliary matrix
AMM
−
1
1
2
(A exists as M
1
and M
2
are positive definite). Let
e
1
and
e
2
be the eigenvectors of A, which may not be orthogonal vectors as A is, in general, not sym-
metric. Metrics M
1
and M
2
are simultaneously diagonalised by the eigenvectors of A such
that
e
=
T
M =
and
e
T
M =
e
112
0
e
122
0
Let
T
e
e
a
0
MQ
a
0
T
1
1
−
1
1
−
1
P
=
ee
,
Q
=
P
=
,
QMP
=
or
=
P
12
1
1
T
0
b
0
b
1
1
2
Similarly,
a
0
MQ
a
0
2
−
1
2
−
1
QM P
=
or
=
P
2
2
0
b
0
b
2
2
The intersection of metrics M
1
and M
2
is another metric given by
MM MQ
a
0
−
1
−
1
=
=
P
here amin aa andb
=
(, )
=
min(, )
b
12
1
2
12
0
b
Let
v
= v
1
e
1
+ v
2
e
2
be a vector expressed in the basis (
e
1
,
e
2
),
T
T
vv e
Mv
=
(
+
v
e
)
Mv
(
e
+
v
e
)
1
1 1
2
2
1
11
22
2
T
2
T
Ma vbv
2
2
=
vM
e
e
+
v
e
2212
e
=+
1
111
2
1
1
12
2
= +
.
Hence,
v
T
M
v
≤
v
T
M
1
v
and
v
T
M
v
≤
v
T
M
2
v
. Thus, M is bounded by M
1
and M
2
.
The intersection of two metrics M
1
and M
2
by simultaneous diagonalisation is shown in
Figure 4.19, in which M
1
= M
1
(θ
1
= 1, λ
1
= 2, μ
1
= 3), M
2
= M
2
(θ
2
= 2 − π/2, λ
2
=5, μ
2
= 1) and
the metric of intersection M
I
= M
I
(θ
I
= 0.3645, λ
I
= 2.2022, μ
I
= 0.9903).
T
M v v
2
2
and
vv
T
Mavbv
2
2
Similarly,
vv
=+
1
2
2
1
22
