“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 229

Distortion energy over a spherical cap: six polar azimuthal projections, 0 ◦ ≤ 60 ◦

Tab l e 5 . 3 Δ ≤

Δ ( ◦ ) 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

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 Δ = 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 ortoaplane 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, Δ )
y ( Λ, Δ )

= r ( Δ ) cos α ( Λ, Δ )

sin α ( Λ, Δ )

or (5.136)

x ( Λ, Δ )
y ( Λ, Δ )

= f ( Δ ) cos g ( Λ, Δ ) ,

sin g ( Λ, Δ )

which are characterized by two functions, namely the radial function f ( Δ )andthe azimuth
function g ( Λ, Δ ). The azimuth function α ( Λ, Δ )= g ( Λ, Δ )is azimuth preserving if α ( Λ, Δ )= Δ .
Accordingly, in general, a pseudo-azimuthal mapping is not azimuth preserving, α ( Λ, Δ ) = Δ .

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.”

Let us bother you with the detailed analysis of distortion for pseudo-azimuthal mappings of
type x = f ( Δ )cos g ( Λ, Δ )and y = f ( Δ )sin g ( Λ, Δ ). In Box 5.21 , we present the left Jacobi
matrix, the left Cauchy-Green matrix and the left principal stretches. In order to supply you
with a visual impression of what is going to happen when you switch from azimuthal to pseudo-
azimuthal, in Fig. 5.29 , we have made an attempt to highlight the images of the coordinate lines
under the following four categories of mapping. (The various categories have been properly chosen
by Tobler ( 1963a ) as long as we intend to map “sphere to plane”.)