F-3 Deformation analysis of vertically/horizontally averaged equiareal cylindric mappings 579
F-3 Deformation analysis of vertically/horizontally averaged equiareal
cylindric mappings
The deformation analysis of vertically and horizontally averaged equiareal cylindric mappings based
upon E. 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 {Λ 1 , Λ 2 } in terms of the invariant tr
C l C
−1
l
,
both for the vertical as well as the horizontal mean of mixed equiareal cylindric mappings of the biaxial
ellipsoid E
2
A,B . In Boxes F.3 and F.4, we present the items of left Cauchy–Green deformation tensor
based upon (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 {G 1 , G 2 } which locally span T E
2
A,B , the tangent space at {Λ, Φ} of E
2
A,B .
In order to analyse the distortion measures in the chart (x, y) ∈ {R
2 , δ αβ } the coordinates of the
right Cauchy–Green deformation tensor for the vertical-horizontal mixed equiareal cylindric mapping
(F.32) and (F.43) in Boxes 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 E
2
A,B are given in
Corollary F.11, in particular, the orientation of the right principal stretches.
Corollary F.8 (Left principal stretches of the vertical–horizontal mean of mixed equiareal cylindric mapping of the biaxial ellipsoid E
2
A,B ).
The left principal stretches {Λ 1 , Λ 2 } of the coordinates c AB of the left Cauchy–Green deformation
tensor normalized with respect to the coordinates G A,B of the left metric tensor are represented by
(F.47) if the mapping equations (F.32) for the vertical mean and (F.43)–(F.46) for the horizontal
mean of mixed equiareal mappings of the biaxial ellipsoid apply.
Λ 1 = +
1
2
tr
C l G
−1
l
+
tr
C l G
−1
l
2 − 4
,
Λ 2 = +
1
2
tr
C l G
−1
l
−
tr
C l G
−1
l
2 − 4
.
(F.47)
First, for the vertical mean holds (F.48).
tr
C l G
−1
l
=
c 11
G 11
+
c 22
G 22
=
=
A
4 (α + β)
4 [A
2 Λ
2 β
2 sin
2 Φ + (αL + βA)
2 ] + (αL + βA)
6
A 2 (α + β) 2 (αL + βA) 4
.
(F.48)
Second, for the horizontal mean holds (F.49).
tr
C l G
−1
l
=
c 11
G 11
+
c 22
G 22
=
=
(αA + βL)
4 + Λ
2 β
2 sin
2 ΦL
2 (αA + βL)
2 + L
4 (α + β)
4
(α + β) 2 L 2 (αA + βL) 2
.
(F.49)
End of Corollary.
F-3 Deformation analysis of vertically/horizontally averaged equiareal
cylindric mappings
The deformation analysis of vertically and horizontally averaged equiareal cylindric mappings based
upon E. 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 {Λ 1 , Λ 2 } in terms of the invariant tr
C l C
−1
l
,
both for the vertical as well as the horizontal mean of mixed equiareal cylindric mappings of the biaxial
ellipsoid E
2
A,B . In Boxes F.3 and F.4, we present the items of left Cauchy–Green deformation tensor
based upon (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 {G 1 , G 2 } which locally span T E
2
A,B , the tangent space at {Λ, Φ} of E
2
A,B .
In order to analyse the distortion measures in the chart (x, y) ∈ {R
2 , δ αβ } the coordinates of the
right Cauchy–Green deformation tensor for the vertical-horizontal mixed equiareal cylindric mapping
(F.32) and (F.43) in Boxes 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 E
2
A,B are given in
Corollary F.11, in particular, the orientation of the right principal stretches.
Corollary F.8 (Left principal stretches of the vertical–horizontal mean of mixed equiareal cylindric mapping of the biaxial ellipsoid E
2
A,B ).
The left principal stretches {Λ 1 , Λ 2 } of the coordinates c AB of the left Cauchy–Green deformation
tensor normalized with respect to the coordinates G A,B of the left metric tensor are represented by
(F.47) if the mapping equations (F.32) for the vertical mean and (F.43)–(F.46) for the horizontal
mean of mixed equiareal mappings of the biaxial ellipsoid apply.
Λ 1 = +
1
2
tr
C l G
−1
l
+
tr
C l G
−1
l
2 − 4
,
Λ 2 = +
1
2
tr
C l G
−1
l
−
tr
C l G
−1
l
2 − 4
.
(F.47)
First, for the vertical mean holds (F.48).
tr
C l G
−1
l
=
c 11
G 11
+
c 22
G 22
=
=
A
4 (α + β)
4 [A
2 Λ
2 β
2 sin
2 Φ + (αL + βA)
2 ] + (αL + βA)
6
A 2 (α + β) 2 (αL + βA) 4
.
(F.48)
Second, for the horizontal mean holds (F.49).
tr
C l G
−1
l
=
c 11
G 11
+
c 22
G 22
=
=
(αA + βL)
4 + Λ
2 β
2 sin
2 ΦL
2 (αA + βL)
2 + L
4 (α + β)
4
(α + β) 2 L 2 (αA + βL) 2
.
(F.49)
End of Corollary.