G-1 The pseudo-cylindrical mapping of the biaxial ellipsoid onto the plane 591
Second, we set up pseudo-cylindrical mapping equations of the biaxial ellipsoid E
2
A 1 ,A 2
onto the
plane R
2 in terms of surface normal ellipsoidal {longitude Λ, latitude Φ}, in particular
x = x(Λ, Φ) = A 1 Λ
cos Φ
1 − E 2 sin
2 Φ
g(Φ) ,
y = y(Φ) = A 1 f (Φ) ,
x :=
x ∈ R
2 ax + by + c = 0
.
(G.8)
The structure of the pseudo-cylindrical mapping equations with unknown functions f (Φ) and g(Φ) is
motivated by the postulate that for g(Φ) = 1 parallel circles of E
2
A 1 ,A 2
should be mapped equidistantly
onto R
2 . Note that for zero relative eccentricity E = 0, we arrive at the pseudo-cylindrical mapping
equations of the sphere S
2
R .
Third, by means of Corollary G.1, we want to show that in the class of pseudo-cylindrical mapping
equations (G.8), only equiareal map projections are possible.
Corollary G.1 (Pseudo-cylindrical mapping equations).
In the class of the pseudo-cylindrical mapping equations of type (G.8), only equiareal map projections
are possible restricting the unknown functions to (G.9). Note that conformal map projections are not
in the class of pseudo-cylindrical mapping equations.
f
= g
−1
or g = f
.
(G.9)
End of Corollary.
For the proof, we are going to construct the left Cauchy–Green deformation tensor and solve its general
eigenvalue problem in order to compute the principal distortions {Λ 1 , Λ 2 }. The tests Λ 1 Λ 2 = 1 for
equiareal and Λ 1 = Λ 2 for conformality are performed.
Proof.
The infinitesimal distance ds between two points x and x + dx, both elements of the plane {R
2 , δ µν }
is by pullback transformed into a {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 .
(G.10)
Throughout, we apply the summation convention over repeated indices, for example, a µ b µ = a 1 b 1 +a 2 b 2
and a i b i = a 1 b 1 +a 2 b 2 +a 3 b 3 . In addition, we adopt the notation (G.11), the symbols for the meridional
radius of curvature M (Φ) and the normal radius of curvature N (Φ), respectively.
M (Φ) :=
A 1 (1 − E
2 )
(1 − E 2 sin
2 Φ) 3/2 ,
N (Φ) :=
A 1
(1 − E 2 sin
2 Φ) 1/2 .
(G.11)
The normal or transverse radius of curvature of E
2
A 1 ,A 2
is the curvature radius of a curve formed by
the intersection of the normal or transverse plane R
2
⊥ T x E
2
A 1 ,A 2
which is normal to the tangent space
T x E
2
A 1 ,A 2
of the ellipsoid E
2
A 1 ,A 2
and is perpendicular to the meridian at a point X ∈ E
2
A 1 ,A 2
.
x = x(Λ, Φ) = N (Φ)Λ cos Φg(Φ) = A 1 ΛP (Φ)g(Φ) ,
y = y(Φ) = A 1 f (Φ) .
(G.12)