1-14 Review: the canonical criteria 91
Box 1.57 (The optimization problem; extremal, left angular shear or right angular shear; maximal angular
distortion).
Optimization problem:
P ±
l =
= arg
˘P
l ∈ [0, 2π]
P
l (ψ l ) = extr.
¯ versus
P ±
r =
= arg
˘P
r ∈ [0, 2π]
P
r (ψ r ) = extr.
¯
,
(1.336)
x := Ψ l , f(x) :=
P
l (Ψ l ) ,
x := Ψ r , f(x) :=
P
r (Ψ r ) ,
a := Λ 1 , b := Λ 2 .
tan f (x) =
tan x
a + b tan
2 x
.
a:= λ 1 , b := λ 2 .
(1.337)
Stationary points:
(tan x)
= 1 + tan
2 x ,
(1.338)
f
(x) = 0
⇔
(tan f (x))
= (1 + tan
2 f (x))f
(x) = 0 ,
(tan f (x))
=
1 + tan
2 x
(a + b tan
2 x) 2 (a − b tan
2 x) ,
(tan f (x))
= 0
⇔
a − b tan
2 x = 0
⇔
tan x = ±
p a
b
,
(1.339)
tan Ψ
±
l = ±
r
Λ 1
Λ 2
versus tan Ψ
±
r = ±
r
λ 1
λ 2
.
(1.340)
Extremal left or right angular shear:
tan
P ±
l = ±
1
2
Λ 1 − Λ 2
√
Λ 1 Λ 2
versus
tan
P ±
r = ±
1
2
λ 1 − λ 2
√
λ 1 λ 2
,
sin x =
tan x
√
1 + tan
2 x
,
sin
P ±
l = ±
Λ 1 − Λ 2
Λ 1 + Λ 2
versus
sin
P ±
r = ±
λ 1 − λ 2
λ 1 + λ 2
.
(1.341)
Maximal angular distortion:
Ω l :=
P +
l −
P −
l = 2
P +
l
versus Ω r :=
P +
r −
P −
r = 2
P +
r ,
Ω l = 2 arcsin
˛
˛
˛
˛
Λ 1 − Λ 2
Λ 1 + Λ 2
˛
˛
˛
˛
versus
Ω r = arcsin
˛
˛
˛
˛
λ 1 − λ 2
λ 1 + λ 2
˛
˛
˛
˛ .
(1.342)