86
1 From Riemann manifolds to Riemann manifolds
Box 1.50 (Local measures for departure of the mapping M
2
l → M
2
r from isometry).
(i) G. B. Airy (1861):
2
lA :=
1
2
ˆ
(Λ 1 − 1)
2 + (Λ 2 − 1)
2 ˜
versus
1
2
ˆ
(λ 1 − 1)
2 + (λ 2 − 1)
2 ˜
=:
2
rA .
(1.312)
(ii) V. V. Kavrajski (1958):
2
lAK :=
1
2
ˆ
(ln Λ 1 )
2 + (ln Λ 2 )
2 ˜
versus
1
2
ˆ
(ln λ 1 )
2 + (ln λ 2 )
2 ˜
=:
2
rAK .
(1.313)
Box 1.51 (Local measures for departure of the mapping M
2
l → M
2
r from equiareal and conformal).
(i) Departure from an equiareal mapping:
2
l areal := (Λ 1 Λ 2 − 1)
2
versus (λ 1 λ 2 − 1)
2 =:
2
r areal .
(1.314)
(ii) Departure from a conformal mapping:
2
l conf := (Λ 1 − Λ 2 )
2
versus (λ 1 − λ 2 )
2 =:
2
r conf .
(1.315)
Box 1.52 (Global measures for departure of the mapping M
2
l → M
2
r from isometry, areomorphism, and
conformeomorphism).
(i) Isometry:
I lA :=
1
S l
Z
dS l
2
lA
versus
1
S r
Z
dS r
2
rA =: I rA ,
I lAK :=
1
S l
Z
dS l
2
lAK
versus
1
S r
Z
dS r
2
rAK =: I rAK .
(1.316)
(ii) Areomorphism:
I l areal :=
1
S l
Z
dS l
2
l areal
versus
1
S r
Z
dS r
2
r areal =: I r areal .
(1.317)
(iii) Conformeomorphism:
I l conf :=
1
S l
Z
dS l
2
l conf
versus
1
S r
Z
dS r
2
r conf =: I r conf .
(1.318)
1-145 Maximal angular distortion
The conformal mapping f : M
2
l → M
2
r had been previously defined by the angular identity Ψ l = Ψ r
or by zero angular shear
l = Ψ l − Ψ r = 0 or
r = Ψ r − Ψ l = 0. By means of the canonical
criteria Λ 1 = Λ 2 or Λ 1 − Λ 2 = 0, we succeeded to formulate an equivalence for conformality. We shall
concentrate here by means of a case study on the deviation of a general mapping f : M
2
l → M
2
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
Lemma 1.20 (Left and right general eigenvalue problem of the Cauchy–Green deformation tensor).
The angular distortion is maximal if Ω l = 2
+
l = 2 arcsin
Λ 1 −Λ 2
Λ 1 +Λ 2
or Ω r = 2
+
r = 2 arcsin
λ 1 −λ 2
λ 1 +λ 2
.
End of Lemma.
1 From Riemann manifolds to Riemann manifolds
Box 1.50 (Local measures for departure of the mapping M
2
l → M
2
r from isometry).
(i) G. B. Airy (1861):
2
lA :=
1
2
ˆ
(Λ 1 − 1)
2 + (Λ 2 − 1)
2 ˜
versus
1
2
ˆ
(λ 1 − 1)
2 + (λ 2 − 1)
2 ˜
=:
2
rA .
(1.312)
(ii) V. V. Kavrajski (1958):
2
lAK :=
1
2
ˆ
(ln Λ 1 )
2 + (ln Λ 2 )
2 ˜
versus
1
2
ˆ
(ln λ 1 )
2 + (ln λ 2 )
2 ˜
=:
2
rAK .
(1.313)
Box 1.51 (Local measures for departure of the mapping M
2
l → M
2
r from equiareal and conformal).
(i) Departure from an equiareal mapping:
2
l areal := (Λ 1 Λ 2 − 1)
2
versus (λ 1 λ 2 − 1)
2 =:
2
r areal .
(1.314)
(ii) Departure from a conformal mapping:
2
l conf := (Λ 1 − Λ 2 )
2
versus (λ 1 − λ 2 )
2 =:
2
r conf .
(1.315)
Box 1.52 (Global measures for departure of the mapping M
2
l → M
2
r from isometry, areomorphism, and
conformeomorphism).
(i) Isometry:
I lA :=
1
S l
Z
dS l
2
lA
versus
1
S r
Z
dS r
2
rA =: I rA ,
I lAK :=
1
S l
Z
dS l
2
lAK
versus
1
S r
Z
dS r
2
rAK =: I rAK .
(1.316)
(ii) Areomorphism:
I l areal :=
1
S l
Z
dS l
2
l areal
versus
1
S r
Z
dS r
2
r areal =: I r areal .
(1.317)
(iii) Conformeomorphism:
I l conf :=
1
S l
Z
dS l
2
l conf
versus
1
S r
Z
dS r
2
r conf =: I r conf .
(1.318)
1-145 Maximal angular distortion
The conformal mapping f : M
2
l → M
2
r had been previously defined by the angular identity Ψ l = Ψ r
or by zero angular shear
l = Ψ l − Ψ r = 0 or
r = Ψ r − Ψ l = 0. By means of the canonical
criteria Λ 1 = Λ 2 or Λ 1 − Λ 2 = 0, we succeeded to formulate an equivalence for conformality. We shall
concentrate here by means of a case study on the deviation of a general mapping f : M
2
l → M
2
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
Lemma 1.20 (Left and right general eigenvalue problem of the Cauchy–Green deformation tensor).
The angular distortion is maximal if Ω l = 2
+
l = 2 arcsin
Λ 1 −Λ 2
Λ 1 +Λ 2
or Ω r = 2
+
r = 2 arcsin
λ 1 −λ 2
λ 1 +λ 2
.
End of Lemma.