202 5 “Sphere to tangential plane”: polar (normal) aspect
Table 5.3. Distortion energy over a spherical cap: six polar azimuthal projections, 0
◦ ≤ ∆ ≤ 60
◦ .
∆[
◦ ]
J 1
J 2
J 3
J 4
J 5
J 6
0
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
5
0.00381 0.00381 0.00761 0.00383 0.00380 0.00095
10
0.01523 0.01531 0.03038 0.01555 0.01508 0.00383
15
0.03427 0.03466 0.06815 0.03591 0.03350 0.00867
20
0.06093 0.06218 0.12063 0.06628 0.05853 0.01555
25
0.09521 0.09830 0.18746 0.10891 0.08944 0.02457
30
0.13713 0.14359 0.26816 0.16728 0.12540 0.03590
35
0.18670 0.19883 0.36222 0.24694 0.16548 0.04971
40
0.24397 0.26495 0.46908 0.35679 0.20872 0.06624
45
0.30899 0.34315 0.58814 0.51184 0.25419 0.08579
50
0.38184 0.43489 0.71882 0.73874 0.30101 0.10872
55
0.46264 0.54198 0.86056 1.08829 0.34843 0.13550
60
0.55155 0.66667 1.01286 1.66667 0.39583 0.16667
5-3 The pseudo-azimuthal projection
Setting up general equations of the mapping “sphere to plane”: the pseudo-azimuthal projection in the
normal aspect (polar aspect).
In a preceding section, we define polar azimuthal projections by the following two postulates. (i) The
images of the circular meridians Λ = constant under an azimuthal mapping are radial straight lines.
(ii) The images of parallel circles Φ = constant or ∆ = constant are concentric circles. Any deviation
from these postulates generates pseudo-azimuthal projections or, in general, mappings of the sphere
S
2
R of radius R to a polar tangential plane T N S
2
R or to a plane P
2
O through the center O of the sphere
S
2
R . Here, we shall only consider general equations of a pseudo-azimuthal mapping of type
x(Λ, ∆)
y(Λ, ∆)
= r(∆)
cos α(Λ, ∆)
sin α(Λ, ∆)
or
x(Λ, ∆)
y(Λ, ∆)
= f (∆)
cos g(Λ, ∆)
sin g(Λ, ∆)
,
(5.136)
which are characterized by two functions, namely the radial function f (∆) and the azimuth function
g(Λ, ∆). The azimuth function α(Λ, ∆) = g(Λ, ∆) is azimuth preserving if α(Λ, ∆) = ∆. Accordingly,
in general, a pseudo-azimuthal mapping is not azimuth preserving, α(Λ, ∆) = ∆.
Question.
Question: “How are the pseudo-azimuthal projections “sphere to plane” classified?” Answer:
“By computing the left Cauchy–Green matrix as well as its left eigenspace.”
Table 5.3. Distortion energy over a spherical cap: six polar azimuthal projections, 0
◦ ≤ ∆ ≤ 60
◦ .
∆[
◦ ]
J 1
J 2
J 3
J 4
J 5
J 6
0
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
5
0.00381 0.00381 0.00761 0.00383 0.00380 0.00095
10
0.01523 0.01531 0.03038 0.01555 0.01508 0.00383
15
0.03427 0.03466 0.06815 0.03591 0.03350 0.00867
20
0.06093 0.06218 0.12063 0.06628 0.05853 0.01555
25
0.09521 0.09830 0.18746 0.10891 0.08944 0.02457
30
0.13713 0.14359 0.26816 0.16728 0.12540 0.03590
35
0.18670 0.19883 0.36222 0.24694 0.16548 0.04971
40
0.24397 0.26495 0.46908 0.35679 0.20872 0.06624
45
0.30899 0.34315 0.58814 0.51184 0.25419 0.08579
50
0.38184 0.43489 0.71882 0.73874 0.30101 0.10872
55
0.46264 0.54198 0.86056 1.08829 0.34843 0.13550
60
0.55155 0.66667 1.01286 1.66667 0.39583 0.16667
5-3 The pseudo-azimuthal projection
Setting up general equations of the mapping “sphere to plane”: the pseudo-azimuthal projection in the
normal aspect (polar aspect).
In a preceding section, we define polar azimuthal projections by the following two postulates. (i) The
images of the circular meridians Λ = constant under an azimuthal mapping are radial straight lines.
(ii) The images of parallel circles Φ = constant or ∆ = constant are concentric circles. Any deviation
from these postulates generates pseudo-azimuthal projections or, in general, mappings of the sphere
S
2
R of radius R to a polar tangential plane T N S
2
R or to a plane P
2
O through the center O of the sphere
S
2
R . Here, we shall only consider general equations of a pseudo-azimuthal mapping of type
x(Λ, ∆)
y(Λ, ∆)
= r(∆)
cos α(Λ, ∆)
sin α(Λ, ∆)
or
x(Λ, ∆)
y(Λ, ∆)
= f (∆)
cos g(Λ, ∆)
sin g(Λ, ∆)
,
(5.136)
which are characterized by two functions, namely the radial function f (∆) and the azimuth function
g(Λ, ∆). The azimuth function α(Λ, ∆) = g(Λ, ∆) is azimuth preserving if α(Λ, ∆) = ∆. Accordingly,
in general, a pseudo-azimuthal mapping is not azimuth preserving, α(Λ, ∆) = ∆.
Question.
Question: “How are the pseudo-azimuthal projections “sphere to plane” classified?” Answer:
“By computing the left Cauchy–Green matrix as well as its left eigenspace.”