“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 231

G l = R 2 sin 2 Δ 0

R 2 , G − l =

,

1
R 2 sin 2 Δ

0

1
R 2

0 0

=

,

C l G − l = c 11 G − 1

f 2 g Λ
R 2 sin 2 Δ

f 2 g Λ g Δ
R 2

11 c 12 G − 1

22

(5.142)

c 12 G − 1

11 c 22 G − 1

f 2 + f 2 g 2 Δ
R 2

f 2 g Λ g Δ
R 2 sin 2 Δ

22

R 2 sin 2 Δ f 2 g Λ + f 2 + f 2 g Δ sin 2 Δ ,

1

tr[C l G − l ]=

R 4 sin 2 Δ f 2 g Λ f 2 + f 2 g Δ −

f 4 g Λ g Δ ,

1

det [C l G − l ]=

(5.143)

R 2 sin 2 Δ f 2 g Λ + g Δ sin 2 Δ + f 2 sin 2 Δ ,

1

tr[C l G − l ]=

1

det [C l G − l ]=

R 4 sin 2 Δ f 2 f 2 g Λ .

Let us comment on the left principal stretches that we have computed in Box 5.21 . First, based on
the parameterized mapping of category B, we have calculated the left Jacobi matrix, namely the
partial derivatives of x ( Λ, Δ )and y ( Λ, Δ ). Second, we succeeded to derive a simple form of the
left Cauchy-Green matrix. Third, the general eigenvalue problem for the matrix pair

{ C l , G l }

Λ 2 G l

leads to the characteristic equation | C l − | = 0. We did not explicitly compute the left

principal stretches . Instead, we took advantage of the invariant representation of the

left eigenspace in terms of the Hilbert invariants J 1 =tr[C l G − l ]and J 2 =det[C l G − l ]. J 1 as well
as J 2 have been explicitly computed.

{ Λ 1 ,Λ 2 }

Question: “Do conformal mappings of the pseudo-azimuthal
type, category B, exist or do equiareal mappings of the
pseudo-azimuthal type, category B, exist?” Answer: “No
conformal mappings of the pseudo-azimuthal type, cate-
gory B, exist, but equiareal indeed do.”

This question may be asked with the left principal stretches Λ 1 and Λ 2 at hand. But how to prove
this answer? Let us prove this answer in two steps.

Proof.
First, we prove the non-existence of a conformal pseudo-azimuthal mapping, category B. The
canonical postulate of conformality, Λ 1 = Λ 2 ,isequivalentto( 5.144 ). The sum of two positive
numbers cannot be zero, in general. The special case g Λ =1and g Δ = 0 transforms the pseudo-
azimuthal mapping, category B, back to the azimuthal mapping, category A.
tr [C l G − l ]=2 det [C l G − l ]; here : ( fg Λ − f sin Δ ) 2 + f 2 g Δ sin 2 Δ =0 . (5.144)

Second, we characterize an equiareal pseudo-azimuthal mapping, category B. The canonical pos-
tulate of an equiareal mapping, Λ 1 Λ 2 =1,isequivalentto( 5.145 ). In consequence, we give an