18 “Sphere to cone”: pseudo-conic projections
Mapping the sphere to a cone: pseudo-conic projections. The Stab–Werner mapping and the Bonne
mapping. Tissot indicatrix.
First, let us develop the general setup of pseudo-conic projections from the sphere to a cone. Second,
let us present special pseudo-conic mappings like the Stab–Werner mapping and the Bonne mapping
including illustrations.
18-1 General setup and distortion measures of pseudo-conic projections
Conic projections, polyconic projections. Mapping equations, deformation tensor. Lemma of Vieta and
postulate of equal area mapping.
In general, pseudo-conic projections are based upon the setting (18.1) if we use spherical longitude Λ
and spherical co-latitude ∆ = π/2 − Φ and polar coordinates α and r. The next extension leaves us
with the polyconic projections of type (18.2). Our analysis is based upon Lemma 18.1.
α = α(Λ, ∆) = Λ cos ∆ , r = r(∆) = f (∆) ,
(18.1)
α(Λ, ∆) = g(∆)Λ cos ∆ , r = r(∆) = f (∆) .
(18.2)
Lemma 18.1 (Equiareal mapping).
A general mapping of any surface to the plane is equiareal or area preserving if and only if
det [C l ] /det [G l ] = 1 .
(18.3)
C l and G l are the left Cauchy–Green matrix and the left metric matrix, respectively.
End of Lemma.
Proof.
det
C l − Λ
2
S G l
= 0
⇔
Λ
4
S − Λ
2
S tr
C l G
−1
l
+ det
C l G
−1
l
= 0 ,
(18.4)
Λ
2
S
+
Λ
2
S
− = 1
(canonical postulate of equal area mapping)
⇔
det [C l ] /det [G l ] = 1
(Lemma of Vieta:
the product of the solutions of a quadratic equation equals the absolute term) .
(18.5)
End of Proof.
From the general form of the deformation tensor, the metric tensor, and the postulate of equal area
mapping, we derive the general structure of equal area pseudo-conic mappings of the sphere in Box 18.1.
As a side remark, we use the result that only pseudo-conic projections of type equal area exist.