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cover the eigenspace of the left Tissot matrix C
l
G
−
1
1
.Dueto
conformality, they are identical,
Λ
1
=
Λ
1
=
Λ
S
.J
l
denotes the left Jacobi map (d
x,
d
y
)
The eigenvalues
{
Λ
1
,Λ
2
}
→
(d
Λ,
d
Φ
), G
r
the matrix of the right metric tensor of the plane generated by developing the
circular cylinder
2
ρA
1
C
of radius
ρA
1
, namely the unit matrix G
r
=I
2
.
End of Lemma.
Definition I.3 (Generalized Mercator projection, Airy optimum).
2
A
1
,A
2
The generalized Mercator projection of the ellipsoid-of-revolution
E
onto the developed cir-
2
ρA
1
cular cylinder
of radius
ρA
1
is called
Airy optimal
if the deviation from an isometry (
I.4
)
in terms of the left principal stretches
C
averaged over a mapping area of interest, namely
the surface integral (
I.5
), is minimal with respect to the unknown dilatation factor
ρ
.
{
Λ
1
,Λ
2
}
1)
2
+(
Λ
2
−
1)
2
(
Λ
1
−
,
(I.4)
2
1
2
S
1)
2
+(
Λ
2
1)
2
]d
S
=min
ρ
J
l
A
:=
[(
Λ
1
−
−
.
(I.5)
area
End of Definition.
is represented by the expression
det[G
l
]d
Λ
d
Φ
,namely
by (
I.6
). In contrast, for the equatorial strip [
Λ
W
,Λ
E
]
2
A
1
,A
2
The infinitesimal surface element of
E
×
[
Φ
S
,Φ
N
] between a longitudinal extension
(
Λ
0
ΔΛ, Λ
0
+
ΔΛ
) and a latitudinal extension (
Φ
S
,Φ
N
), the finite area is computed by (
I.7
); the
subscripts S, N, E, and W denote South, North, East, and West. The actual computation up to the
fourth order in relative eccentricity, namely O(
E
4
), is performed by an uniform convergent series
expansion of (1
−
x
)
−
2
for
<
1 and a term-wise integration, namely interchanging summation
and integration. Note that along the surface normal longitude of reference
Λ
0
the strip has been
chosen symmetrically such that (
I.8
), i.e.
Λ
E
−
|
x
|
−
Λ
W
=
Λ
0
+
ΔΛ
−
(
Λ
0
−
ΔΛ
)=2
ΔΛ
,holds.
2
The areal element of
A
1
,A
2
is provided by
d
S
=
A
1
(1
−
E
2
)cos
Φ
(1
E
E
2
sin
2
Φ
)
d
Λ
d
Φ,
(I.6)
−
=
Λ
E
Λ
W
d
Λ
Φ
N
Φ
S
S
:= area
A
1
(1
E
2
)cos
Φ
(1
− E
2
sin
2
Φ
)
2
d
Φ
=
−
2
Λ
W
=
Λ
0
−ΔΛ≤Λ≤Λ
ε
=
Λ
0
−ΔΛ
Φ
S
≤Φ≤Φ
N
E
A
1
,A
2
|
=
A
1
(1
− E
)
Λ
E
Λ
W
d
Λ
Φ
N
Φ
S
cos
Φ
[1 + 2
E
2
sin
2
Φ
+O(
E
4
)]d
Φ
=
(I.7)
=2
A
1
(1
− E
2
)
ΔΛ
[sin
Φ
N
+
2
3
E
2
sin
3
Φ
N
−
(sin
Φ
S
+
2
3
E
2
sin
3
Φ
S
)+O(
E
4
)]
,
Λ
E
−
Λ
W
=2
ΔΛ.
(I.8)
Lemma I.4 (Generalized Mercator projection, Airy distortion energy).
In case of the generalized Mercator projection, the left
Airy distortion energy J
l
A
is the quadratic
form in terms of the dilatation factor
ρ
, in particular
2
c
1
ρ
+
c
2
ρ
2
,
J
l
A
(
ρ
)=
c
0
−
(I.9)
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