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+O(
E
4
)
.
+
E
2
(
α
−
β
cos
Φ
)(2
α
+3
β
cos
Φ
)sin
Φ
4
β
2
(
α
+
β
cos
Φ
)
α<β
:
E
2
)(
α
+
β
)
y
=
y
(
Φ
)
≈
A
(1
−
×
α
2
ln
β
+
α
cos
Φ
+
β
2
1+
E
2
1
2
β
2
+
− α
2
sin
Φ
α
+
β
cos
Φ
α
2
α
β
β
2
×
−
−
−
1+
E
2
3
2
β
2
+
α
2
+
Φ
β
4
−
(F.45)
+O(
E
4
)
.
+
E
2
(
α
−
β
cos
Φ
)(2
α
+3
β
cos
Φ
)sin
Φ
4
β
2
(
α
+
β
cos
Φ
)
α
=
β
=1:
2
AΛ
1
E
2
sin
2
Φ
+c
os
Φ
1
−
1
x
=
x
(
Λ, Φ
)
≈
,
E
2
sin
2
Φ
−
y
=
y
(
Φ
)
≈
2
A
(1
− E
2
)
×
(F.46)
Φ
1+
E
2
4
1+
E
2
2
+
E
2
1
4(1 + cos
Φ
)
(2 + 3 cos
Φ
)sin
Φ
+O(
E
2
)
.
tan
Φ
2
−
cos
Φ
×
−
End of Lemma.
F-3 Deformation Analysis of Vertically/Horizontally Averaged
Equiareal Cylindric Mappings
The deformation analysis of vertically and horizontally averaged equiareal cylindric mappings
based upon
Grafarend
(
1995
) will inform us about the minimal and maximal scale distortions,
also called
left
and
right principal stretches
, as well as the maximal angular distortion. Indeed,
we collect in Corollary
F.8
the representation of left principal stretches
in terms of
the invariant tr [C
l
C
−
l
], both for the vertical as well as the horizontal mean of mixed equiareal
cylindric mappings of the biaxial ellipsoid
{
Λ
1
,Λ
2
}
2
A,B
.InBoxes
F.3
and
F.4
, we present the items of
left
Cauchy-Green deformation tensor
basedupon(
F.32
)or(
F.43
)-(
F.46
). Corollary
F.9
, in contrast,
reviews the representation of maximal
left angular distortion
in terms of the sum and the difference
of the
left principal stretches
. The two left eigenvectors of the left Cauchy-Green deformation are
by Corollary
F.10
computed in the basis of images of tangent vectors
E
{
G
1
,G
2
}
which locally
2
2
span
T
E
A,B
, the tangent space at
{
Λ, Φ
}
of
E
Λ,B
. In order to analyse the distortion measures
2
,δ
αβ
in the chart (
x, y
)
the coordinates of the
right Cauchy-Green deformation tensor
for the vertical-horizontal mixed equiareal cylindric mapping (
F.32
)and(
F.43
)inBoxes
F.5
and
F.6
have been computed. The right principal stretches are inversely related to the left principal
stretches. The right eigenvectors of the right Cauchy-Green deformation tensor of the vertical-
horizontal mean of mixed equiareal cylindric mapping of
∈{
R
}
2
A,B
are given in Corollary
F.11
,in
E
particular, the orientation of the right principal stretches.
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