Geography Reference
In-Depth Information
Fig. 18.3.
Bonne mapping, pseudo-conic projection, Tissot ellipses of distortion
Exercise 18.1 (Stab-Werner projection).
In the sixteenth and seventeenth centuries, the pseudo-conic, the equal area, and the cordiform
(heart shaped)
Stab-Werner projections
(J. Stab and J. Werner
1514) was frequently used for
world maps
and some
continental maps
. The mapping equations are specified through polar coor-
dinates
α
and
r
or Cartesian coordinates
x
and
y
as follows. (
Λ
and
Φ
describe spherical longitude
and spherical latitude.)
∼
cos
Φ
π/
2
α
=
Λ
Φ
, r
=
R
(
π/
2
−
Φ
)
,
(18.34)
−
x
=
R
(
π/
2
− Φ
)cos
Λ
, y
=
R
(
π/
2
− Φ
)sin
Λ
.
cos
Φ
π/
2
cos
Φ
π/
2
(18.35)
−
Φ
−
Φ
(i) Prove analytically that the Stab-Werner projection is equal area and (ii) determine the numer-
ical values of the elements of the Tissot indicatrix (Tissot ellipse: minor distortions and major
distortions, and coordinates of the corresponding eigendirections in the map) for the point
Aachen,
Germany
(
Λ
=6
◦
06
E,
Φ
=50
◦
46
N).
Solution.
From the general eigenvalue problem in Lemma
1.7
or from (
18.26
), the minor and the major
distortions are easily calculated as (
18.36
), and det D
l
=
Λ
min
Λ
max
= 1. The matrix F
l
of eigen-
vectors, fulfilling the requirements F
l
C
l
F
l
=D
l
and F
l
G
l
F
l
=I
2
(“left diagonalization”), results
to (
18.37
). These eigenvectors refer to the base vectors of the left tangential space
T
M
l
.Inorder
to properly plot the Tissot ellipses of distortion, the eigendirections in the right tangential space
T
M
r
are needed.
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