F-1 Pseudo-cylindrical mapping: biaxial ellipsoid onto plane 571
Corollary F.1 (Pseudo-cylindrical mapping equations).
In the class of pseudo-cylindrical mapping equations of type (F.1), only equiareal map projections are
possible restricting the unknown functions to (F.8). Conformal map projections are not in the class
of pseudo-cylindrical mapping equations.
g = Mf
or f
= Mg
−1 .
(F.8)
End of Corollary.
For the proof, we are going to construct the left Cauchy–Green deformation tensor according to
E. Grafarend (1995 pp. 432–436) and solve its general eigenvalue problem with respect to the metric
matrix of the biaxial ellipsoid E
2
A,B , i. e. in order to compute the left principal stretches {Λ 1 , Λ 2 }.
The tests Λ 1 Λ 2 = 1 for an equiareal mapping and Λ 1 = Λ 2 for a conformal mapping according to
E. Grafarend (1995 p. 449) are performed.
Proof.
The infinitesimal distance ds between two points x and x + dx both elements of the plane {R
2 , δ µν }
is pullback transformed into the {Λ, Φ} coordinate representation, in particular
ds
2 = dx
2 + dy
2 = δ µν dx
µ dx
ν =
= δ µν
∂x
µ
∂U A
∂x
ν
∂U B dU
A dU
B
∀ U
1 = Λ, U
2 = Φ ,
ds
2 = c AB dU
A dU
B
∀ c AB := δ µν
∂x
µ
∂U A
∂x
ν
∂U B .
(F.9)
Throughout, we apply the summation convention over repeated indices. For example, we here apply
a µ b µ = a 1 b 1 + a 2 b 2 . In addition, we adopt the symbols provided by (F.10) as the symbols for the
meridional radius of curvature M (Φ) and the normal radius of curvature N (Φ), respectively.
M (Φ) :=
A(1 − E
2 )
(1 − E 2 sin
2 Φ) 3/2 , N(Φ) :=
A
(1 − E 2 sin
2 Φ) 1/2 .
(F.10)
Indeed, M (Φ) is the radius of curvature of the meridian, the coordinate line Λ = const., but N (Φ) as
the transverse radius of curvature of a curve formed by the intersection of the normal or transverse
plane P
2
⊥ T x E
2
A,B which is normal to the tangent space T x E
2
A,B of the biaxial ellipsoid E
2
A,B and is
perpendicular to the meridian at a point {Λ, Φ} ∈ E
2
A,B . In contrast, N (Φ) cos Φ = L(Φ) is the radius
of curvature of the parallel circle, the coordinate line Φ = const. The principal curvature radii of the
biaxial ellipsoid E
2
A,B are {M (Φ), N(Φ)}.
x = x(Λ, Φ) = [N (Φ) cos Φ]Λg(Φ) = L(Φ)Λg(Φ) ,
y = y(Φ) = f (Φ) .
(F.11)
The left Cauchy–Green deformation tensor c AB is generated by (F.12). (x Λ , y Λ , x Φ , y Φ ) denote the
partials of (x, y) with respect to (Λ, Φ) so that we are led to (F.13).
c 11 = x
2
Λ + y
2
Λ , c 12 = x Λ x Φ + y Λ y Φ , c 21 = c 12 , c 22 = x
2
Φ + y
2
Φ ,
(F.12)
{c AB } =
L
2 g
2
ΛLg(L
g + Lg
)
ΛLg(L
g + Lg
) Λ
2 (L
g + Lg
)
2 + f
2
.
(F.13)