5-4 The Wiechel polar pseudo-azimuthal projection 205
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 H. 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, ∆) = Λ (in general, here we
consider α = Λ + ∆/2), the Wiechel map is pseudo-azimuthal. A quick view to Wiechel’s pseudoazimuthal 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, ∆) = e 1 x(Λ, ∆) + e 2 y(Λ, ∆):
curl x(Λ, ∆) = e 3 (D Λ y − D ∆ x) = e 3
−R cos(Λ + ∆) + 2R sin
∆
2
cos
Λ +
∆
2
= 0 . (5.146)
Consult the original contribution of H. 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.
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 ∆ x, D Λ y, and D ∆ 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
−1
l , namely Λ 1 and Λ 2 , from the trace tr
C l G
−1
l
and the determinant
det
C l G
−1
l
. Fourth, Λ 1 Λ 2 = 1 proves an equiareal mapping.
Fig. 5.30. Wiechel’s pseudo-azimuthal projection “sphere to plane”, normal aspect.