402 18 “Sphere to cone”: pseudo-conic projections
Exercise 18.1 (Stab–Werner projection).
In the 16th century and the 17th century, 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 trough polar coordinates α and r or
Cartesian coordinates x and y as follows. (Λ and Φ describe spherical longitude and spherical latitude.)
α = Λ
cos Φ
π/2 − Φ
, r = R(π/2 − Φ) ,
(18.34)
x = R(π/2 − Φ) cos
Λ
cos Φ
π/2 − Φ
, y = R(π/2 − Φ) sin
Λ
cos Φ
π/2 − Φ
.
(18.35)
(i) Prove analytically that the Stab–Werner projection is equal area and (ii) determine the numerical
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 eigenvectors, fulfilling
the requirements F
T
l C l F l = D
2
l and F
T
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
2
l . In order to properly plot the
Tissot ellipses of distortion, the eigendirections in the right tangential space T M
2
r are needed.
D l :=
:= diag [Λ min , Λ max ] =
0.992 10
0
0
1.007 97
,
(18.36)
F l =
=
1.122 42 1.113 55
−0.704 30 0.709 91
.
(18.37)
With the help of the left Jacobi matrix
J l =
D Λ x D Φ x
D Λ y D Φ y
Λ=6 ◦ 06 E, Φ=50 ◦ 46 N
=
−0.062 10 −0.996 73
0.629 42 −0.082 38
,
(18.38)
the transformation left-to-right is easily performed as
F r = J l F l D
−1
l
= J l F l D r =
0.770 59 0.637 33
−0.637 33 0.770 59
,
(18.39)
and
F
T
r C r F r = D
2
r , F
T
r G r F r = F
T
r F r = I 2 .
(18.40)
Indeed, F r is an orthonormal matrix.
End of Exercise.
Exercise 18.1 (Stab–Werner projection).
In the 16th century and the 17th century, 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 trough polar coordinates α and r or
Cartesian coordinates x and y as follows. (Λ and Φ describe spherical longitude and spherical latitude.)
α = Λ
cos Φ
π/2 − Φ
, r = R(π/2 − Φ) ,
(18.34)
x = R(π/2 − Φ) cos
Λ
cos Φ
π/2 − Φ
, y = R(π/2 − Φ) sin
Λ
cos Φ
π/2 − Φ
.
(18.35)
(i) Prove analytically that the Stab–Werner projection is equal area and (ii) determine the numerical
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 eigenvectors, fulfilling
the requirements F
T
l C l F l = D
2
l and F
T
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
2
l . In order to properly plot the
Tissot ellipses of distortion, the eigendirections in the right tangential space T M
2
r are needed.
D l :=
:= diag [Λ min , Λ max ] =
0.992 10
0
0
1.007 97
,
(18.36)
F l =
=
1.122 42 1.113 55
−0.704 30 0.709 91
.
(18.37)
With the help of the left Jacobi matrix
J l =
D Λ x D Φ x
D Λ y D Φ y
Λ=6 ◦ 06 E, Φ=50 ◦ 46 N
=
−0.062 10 −0.996 73
0.629 42 −0.082 38
,
(18.38)
the transformation left-to-right is easily performed as
F r = J l F l D
−1
l
= J l F l D r =
0.770 59 0.637 33
−0.637 33 0.770 59
,
(18.39)
and
F
T
r C r F r = D
2
r , F
T
r G r F r = F
T
r F r = I 2 .
(18.40)
Indeed, F r is an orthonormal matrix.
End of Exercise.