Geography Reference
In-Depth Information
The right eigencolumns, which are here denoted as (
v
1
,
v
2
), are constructed from the following
system of equations:
−
27
.
329
v
11
+3
v
21
=0
,
+0
.
329
v
12
+3
v
22
=0
,
(1.42)
v
11
+
v
21
=1
,v
12
+
v
22
=1
.
This system of equations leads to two solutions. In the frame of the example to be considered
here, we have chosen the following result:
v
11
=+0
.
109 117
,v
12
=+0
.
994 029
,
(1.43)
v
21
=+0
.
994 029
,v
22
=
−
0
.
109 117
.
In summary, the
left and right eigencolumns
are collected in the two following orthonormal matri-
ces U and V:
U=
+0
.
346 946 + 0
.
937 665
,
V=
+0
.
109 117 + 0
.
994 029
.
(1.44)
+0
.
937 665
−
0
.
346 946
+0
.
994 029
−
0
.
109 117
The
polar decomposition
is now straightforward. According to the above considerations, we finally
arrive at the result
R=UV
∗
,
S=VΣV
∗
,
Σ=diag(
σ
1
,σ
2
)
,
(1.45)
R=
+0
.
970 142 + 0
.
242 536
,
S=
+5
.
093 248 + 0
.
242 536
.
(1.46)
−
0
.
242 536 + 0
.
970 142
+0
.
242 536 + 7
.
276 069
Note that from this result immediately follows that R is an orthonormal matrix. Furthermore,
note that S indeed is a symmetric matrix.
End of Example.
Before we consider a second multiplicative measure of deformation, please enjoy Fig.
1.5
,which
shows the
Hammer retroazimuthal projection
, illustrating special mapping equations of the sphere.
The
ID card
of this special pseudo-azimuthal map projection is shown in Table
1.1
.
1-2 Stretch or Length Distortion
A second multiplicative measure of deformation: stretch or length distortion, Tissot portrait,
simultaneous diagonalization of two matrices.
The
second multiplicative measure
of deformation is based upon the scale ratio, which is also called
stretch, dilatation factor
,or
length distortion
. One here distinguishes the left and right stretch:
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