582 F Mixed cylindric map projections
Corollary F.9 (Maximal left angular distortion of the vertical–horizontal mean of mixed equiareal cylindric
mapping of the biaxial ellipsoid E
2
A,B ).
The right maximal angular distortion Ω generated by 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
is represented by (F.62) subject to (F.63), where tr
C l G
−1
l
is given by (F.48) for the vertical mean
of (F.32) and by (F.49) for the horizontal mean of (F.43)–(F.46).
Ω = arc sin
Λ 1 − Λ 2
Λ 1 + Λ 2
,
(F.62)
Λ 1 − Λ 2 =
tr
C l G
−1
l
− 2 , Λ 1 + Λ 2 =
tr
C l G
−1
l
+ 2 .
(F.63)
End of Corollary.
Proof.
For the proof of (F.63) just compute (Λ 1 − Λ 2 )
2 and (Λ 1 + Λ 2 )
2 under the postulate of an equiareal
mapping Λ 1 Λ 2 = 1, namely by means of (F.48).
(Λ 1 − Λ 2 )
2 = Λ
2
1 + Λ
2
2 − 2Λ 1 Λ 2 = Λ
2
1 + Λ
2
2 − 2 = tr
C l G
−1
l
− 2 ,
(Λ 1 + Λ 2 )
2 = Λ
2
1 + Λ
2
2 + 2Λ 1 Λ 2 = Λ
2
1 + Λ
2
2 + 2 = tr
C l G
−1
l
+ 2 .
(F.64)
End of Proof.
Corollary F.10 (The left eigenvectors of the left Cauchy–Green deformation tensor).
The left eigenvectors of the left Cauchy–Green deformation tensor normalized with respect to the left
metric tensor can be represented with respect to the basis {G 1 , G 2 } which spans the local tangent
space T E
2
A,B
End of Corollary.
The infinitesimal distance dS between two points X and X +dX both elements of the biaxial ellipsoid
E
2
A,B , see (F.65), is push–forward transformed into the {x, y} coordinate representation, in particular,
into (F.66) and (F.67).
f l1 =
−(c 12 − Λ
2
1 G 12 )G 2 + (c 22 − Λ
2
1 G 22 )G 2
G 22 (c 12 − Λ 2
1 G 12 ) 2 + G 11 (c 22 − Λ 2
1 G 22 ) 2 − 2G 12 (c 12 − Λ 2
1 G 12 )(c 22 − Λ 2
1 G 22 )
,
f l2 =
−(c 12 − Λ
2
2 G 12 )G 1 + (c 11 − Λ
2
2 G 11 )G 2
G 11 (c 12 − Λ 2
2 G 12 ) 2 + G 22 (c 11 − Λ 2
2 G 11 ) 2 − 2G 12 (c 12 − Λ 2
2 G 12 )(c 11 − Λ 2
2 G 11 )
,
(F.65)
dS
2 = G AB dU
A dU
B = G AB
∂U
A
∂u α
∂U
B
∂u β du
α du
β
∀ u
1 = x , u
2 = y ,
(F.66)
dS
2 = C αβ du
α du
β
∀ C αβ := G AB
∂U
A
∂u α
∂U
B
∂u β .
(F.67)
Corollary F.9 (Maximal left angular distortion of the vertical–horizontal mean of mixed equiareal cylindric
mapping of the biaxial ellipsoid E
2
A,B ).
The right maximal angular distortion Ω generated by 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
is represented by (F.62) subject to (F.63), where tr
C l G
−1
l
is given by (F.48) for the vertical mean
of (F.32) and by (F.49) for the horizontal mean of (F.43)–(F.46).
Ω = arc sin
Λ 1 − Λ 2
Λ 1 + Λ 2
,
(F.62)
Λ 1 − Λ 2 =
tr
C l G
−1
l
− 2 , Λ 1 + Λ 2 =
tr
C l G
−1
l
+ 2 .
(F.63)
End of Corollary.
Proof.
For the proof of (F.63) just compute (Λ 1 − Λ 2 )
2 and (Λ 1 + Λ 2 )
2 under the postulate of an equiareal
mapping Λ 1 Λ 2 = 1, namely by means of (F.48).
(Λ 1 − Λ 2 )
2 = Λ
2
1 + Λ
2
2 − 2Λ 1 Λ 2 = Λ
2
1 + Λ
2
2 − 2 = tr
C l G
−1
l
− 2 ,
(Λ 1 + Λ 2 )
2 = Λ
2
1 + Λ
2
2 + 2Λ 1 Λ 2 = Λ
2
1 + Λ
2
2 + 2 = tr
C l G
−1
l
+ 2 .
(F.64)
End of Proof.
Corollary F.10 (The left eigenvectors of the left Cauchy–Green deformation tensor).
The left eigenvectors of the left Cauchy–Green deformation tensor normalized with respect to the left
metric tensor can be represented with respect to the basis {G 1 , G 2 } which spans the local tangent
space T E
2
A,B
End of Corollary.
The infinitesimal distance dS between two points X and X +dX both elements of the biaxial ellipsoid
E
2
A,B , see (F.65), is push–forward transformed into the {x, y} coordinate representation, in particular,
into (F.66) and (F.67).
f l1 =
−(c 12 − Λ
2
1 G 12 )G 2 + (c 22 − Λ
2
1 G 22 )G 2
G 22 (c 12 − Λ 2
1 G 12 ) 2 + G 11 (c 22 − Λ 2
1 G 22 ) 2 − 2G 12 (c 12 − Λ 2
1 G 12 )(c 22 − Λ 2
1 G 22 )
,
f l2 =
−(c 12 − Λ
2
2 G 12 )G 1 + (c 11 − Λ
2
2 G 11 )G 2
G 11 (c 12 − Λ 2
2 G 12 ) 2 + G 22 (c 11 − Λ 2
2 G 11 ) 2 − 2G 12 (c 12 − Λ 2
2 G 12 )(c 11 − Λ 2
2 G 11 )
,
(F.65)
dS
2 = G AB dU
A dU
B = G AB
∂U
A
∂u α
∂U
B
∂u β du
α du
β
∀ u
1 = x , u
2 = y ,
(F.66)
dS
2 = C αβ du
α du
β
∀ C αβ := G AB
∂U
A
∂u α
∂U
B
∂u β .
(F.67)