Geography Reference
In-Depth Information
2
l
2
Box 1.52 (Global measures for departure of the mapping
M
→
M
r
from isometry, areomor-
phism, and conformeomorphism).
(i) Isometry:
d
S
l
l
A
versus
d
S
r
rA
=:
I
rA
,
I
l
A
:=
1
1
S
r
(1.316)
S
l
d
S
l
l
AK
versus
d
S
r
rAK
=:
I
rAK
.
I
l
AK
:=
1
S
l
1
S
r
(ii) Areomorphism:
d
S
l
l
areal
versus
d
S
r
rareal
=:
I
rareal
.
I
l
areal
:=
1
1
S
r
(1.317)
S
l
(iii) Conformeomorphism:
d
S
l
l
conf
versus
d
S
r
rconf
=:
I
rconf
.
I
l
conf
:=
1
S
l
1
S
r
(1.318)
1-145 Maximal Angular Distortion
2
l
2
The conformal mapping
f
:
M
→
M
r
had been previously defined by the
angular identity Ψ
l
=
Ψ
r
or by
zero angular shear
l
=
Ψ
l
−
Ψ
r
=0or
r
=
Ψ
r
−
Ψ
l
=0.Bymeansofthe
canonical criteria
Λ
1
=
Λ
2
or
Λ
1
−
Λ
2
= 0, we succeeded to formulate an equivalence for conform
al
ity. We shall
concentrate here by means of a case study on the deviation of a general mapping
f
:
I
r
from conformality. In particular, we shall solve the
optimization problem
of
maximal angular
shear
or of the largest deviation of such a general mapping from conformality. Fast first-hand
information is offered by Lemma
1.20
2
l
M
→
M
Lemma 1.20 (Left and right general eigenvalue problem of the Cauchy-Green deformation ten-
sor).
The angular distortion is maximal if
Ω
l
=2
l
=2arcsin
Λ
1
−
Λ
2
or
Ω
r
=2
r
=2arcsin
λ
1
−
λ
2
.
Λ
1
+
Λ
2
λ
1
+
λ
2
End of Lemma.
The general proof of such a lemma can be taken from
Truesdell and Toupin
(
1960
), pp. 257-266.
here, we make the simplifying assumption
{G
12
=0
,c
12
=0
}
and
{g
12
=0
,C
12
=0
}
.Theoff-
diagonal elements of the left matrix of the metric G
l
as well as of the left Cauchy-Green matrix
C
l
vanish. Or we may say that the coordinate lines “left” and their images “right” intersect at
right angles. In consequence, the mapping equations are specified by
{u
(
U
)
,v
(
V
)
}
. An analogue
statement can be made for the special case
{g
12
=0
,C
12
=0
}
.First,wehavetodefinethe
angular parameters
Ψ
l
and
Ψ
r
.AccordingtoFigs.
1.30
and
1.31
, we refer the angle
Ψ
l
and
Ψ
r
,
respectively, to the unit tangent vector
C
1
along the
V
= constant coordinate line and to the
unit tangent vector
D
1
of an arbitrary curve intersecting the coordinate line
V
= constant, as
well as to the unit tangent vector
c
1
along the
v
= constant coordinate line and to the unit
tangent vector
d
1
of an arbitrary curve intersecting the coordinate line
v
= constant. Such an
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