10
1 From Riemann manifolds to Riemann manifolds
This system of equations leads to two solutions. In the frame of the example to be considered here,
we have chosen the following result:
u 11 = +0.346 946 , u 12 = +0.937 885 ,
u 21 = +0.937 885 , u 22 = −0.346 946 .
(1.41)
The right eigencolumns, which are here denoted as (v 1 , v 2 ), are constructed from the following system
of equations:
−27.329v 11 + 3v 21 = 0 , +0.329v 12 + 3v 22 = 0 ,
v
2
11 + v
2
21 = 1 , v
2
12 + v
2
22 = 1 .
(1.42)
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 ,
v 21 = +0.994 029 , v 22 = −0.109 117 .
(1.43)
In summary, the left and right eigencolumns are collected in the two following orthonormal matrices
U and V:
U =
+0.346 946 +0.937 665
+0.937 665 −0.346 946
, V =
+0.109 117 +0.994 029
+0.994 029 −0.109 117
.
(1.44)
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
−0.242 536 +0.970 142
, S =
+5.093 248 +0.242 536
+0.242 536 +7.276 069
.
(1.46)
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.
Table 1.1. ID card of Hammer retroazimuthal projection of the sphere.
(i) Classification
Retroazimuthal, modified azimuthal, neither conformal nor equal area.
(ii) Graticule
Meridians: central meridian is straight, other meridians are curved.
Parallels: curved. Poles of the sphere: curved lines.
Symmetry: about the central meridians.
(iii) Distortions
Distortions of area and shape
(iv) Other features
The direction from any point to the center of the map is the angle that a
straight line connecting the two points makes with a vertical line. This
feature is the basis of the term “retroazimuthal”. Scimitar-shaped
boundary. Considerable overlapping when entire sphere is shown.
(v) Usage
To determine the direction of a central point from a given location
(vi) Origins
Presented by E. Hammer (1858–1925) in 1910. The author is the successor
of E. Hammer in the Geodesy Chair of Stuttgart University (Germany). The
map projection was independently presented by E. A. Reeves (1862–1945)
and A. R. Hinks (1874–1945) of England in 1929.
1 From Riemann manifolds to Riemann manifolds
This system of equations leads to two solutions. In the frame of the example to be considered here,
we have chosen the following result:
u 11 = +0.346 946 , u 12 = +0.937 885 ,
u 21 = +0.937 885 , u 22 = −0.346 946 .
(1.41)
The right eigencolumns, which are here denoted as (v 1 , v 2 ), are constructed from the following system
of equations:
−27.329v 11 + 3v 21 = 0 , +0.329v 12 + 3v 22 = 0 ,
v
2
11 + v
2
21 = 1 , v
2
12 + v
2
22 = 1 .
(1.42)
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 ,
v 21 = +0.994 029 , v 22 = −0.109 117 .
(1.43)
In summary, the left and right eigencolumns are collected in the two following orthonormal matrices
U and V:
U =
+0.346 946 +0.937 665
+0.937 665 −0.346 946
, V =
+0.109 117 +0.994 029
+0.994 029 −0.109 117
.
(1.44)
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
−0.242 536 +0.970 142
, S =
+5.093 248 +0.242 536
+0.242 536 +7.276 069
.
(1.46)
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.
Table 1.1. ID card of Hammer retroazimuthal projection of the sphere.
(i) Classification
Retroazimuthal, modified azimuthal, neither conformal nor equal area.
(ii) Graticule
Meridians: central meridian is straight, other meridians are curved.
Parallels: curved. Poles of the sphere: curved lines.
Symmetry: about the central meridians.
(iii) Distortions
Distortions of area and shape
(iv) Other features
The direction from any point to the center of the map is the angle that a
straight line connecting the two points makes with a vertical line. This
feature is the basis of the term “retroazimuthal”. Scimitar-shaped
boundary. Considerable overlapping when entire sphere is shown.
(v) Usage
To determine the direction of a central point from a given location
(vi) Origins
Presented by E. Hammer (1858–1925) in 1910. The author is the successor
of E. Hammer in the Geodesy Chair of Stuttgart University (Germany). The
map projection was independently presented by E. A. Reeves (1862–1945)
and A. R. Hinks (1874–1945) of England in 1929.