“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 232

example of an equiareal pseudo-azimuthal mapping, category B. The special case g Λ = 1 trans-
forms the pseudo-azimuthal mapping, category B, back to the azimuthal mapping, category A.
det [C l G − l ]=1; here:

ff
R 2 sin Δ g Λ =1 .

(5.145)

End of Proof.

5-4 The Wiechel Polar Pseudo-Azimuthal Projection

A special variant of Lambert's equiareal polar azimuthal projection: the Wiechel polar
pseudo-azimuthal projection.

A special variant of Lambert's equiareal polar azimuthal projection has been given by
Wiechel ( 1879 ). The direct equations for mapping the “sphere to plane” are presented in Box 5.22
and are illustrated in Fig. 5.30 . Thanks to the azimuthal function α = g ( Λ, Δ ) = Λ (in general,

here we consider α = Λ + Δ/ 2), the Wiechel map is pseudo-azimuthal. A quick view to Wiechel's
pseudo-azimuthal map of Fig. 5.30 motivates the following interpretation: we see the polar vortex
at the North Pole directed to the Earth's rotation axis, namely e 3 . Indeed, we compute the curl
or vortex of the placement vector x ( Λ, Δ )= e 1 x ( Λ, Δ )+ e 2 y ( Λ, Δ ):

curl x ( Λ, Δ )= e 3 ( D Λ y − D Δ x )

= e 3

2 cos Λ + Δ

R cos( Λ + Δ )+2 R sin Δ

− =0 . (5.146)

2

Consult the original contribution of Wiechel ( 1879 ) for a deeper understanding. In particular,
enjoy his arguments for “a rotational graticule”. To become familiar with such a special pseudo-
azimuthal mapping “sphere to plane”, let us ask the following question.

Question: “Is the Wiechel pseudo-azimuthal projection
“sphere to plane” equiareal?” Answer: “Yes.”

For the proof, follow the lines of the proof outlined in Box 5.22 . First, we compute the left Jacobi
matrix constituted by the partial derivatives D Λ x, D Δ x, D Λ y ,and D Δ y . Second, we derive the
left Cauchy-Green matrix by computing C l =J l J l . Third, we derive the left principal stretches,

the left eigenvalues of the matrix C l G − l ,namely Λ 1 and Λ 2 , from the trace tr C l G − l and the

determinant det C l G − l . Fourth, Λ 1 Λ 2 = 1 proves an equiareal mapping.