Geography Reference
In-Depth Information
Box 1.53 (Reference frames (3-leg) of type Gauss, Cartan, and Darboux).
1
l
2
The left manifolds (
M
⊂
M
l
,G
12
=0)
.
Cartan :
C
1
:=
Darboux :
D
1
;=
X
=
d
d
S
,
D
2
;=
D
3
Gauss :
G
1
:=
∂
X
(
U,V
)
G
1
G
1
G
1
√
G
11
=
,
∂U
,
G
2
:=
∂
X
(
U,V
)
G
2
G
2
G
2
C
2
:=
=
,
×
D
1
=
(1.322)
√
G
22
,
=
D
1
)
,
D
3
=
C
3
=
G
3
.
∗
(
D
3
∧
∂V
C
3
:=
C
1
×
C
2
=
G
1
×
G
2
G
1
×
G
2
G
3
:=
.
=
∗
(
C
1
∧
C
2
)=
G
3
.
The right manifolds (
c
12
=0)
.
Cartan :
c
1
:=
Darboux :
d
1
:=
x
=
d
x
(
u
(
s
)
,v
(6))
Gauss :
g
1
:=
∂
x
(
u
(
U
)
,v
(
V
))
g
1
g
1
,
,
,
d
s
g
2
∂U
c
2
:=
,
c
3
:=
c
1
×
c
2
=
=
∗
(
c
1
∧
c
2
)
.
d
2
:=
d
3
×
d
1
=
=
∗
(
d
3
∧ d
1
)
,
d
3
=
c
3
=
g
3
.
(1.323)
g
2
:=
∂
x
(
u
(
U
)
,v
(
V
))
,
g
2
∂V
g
1
×
g
2
g
1
×
g
2
g
3
:=
.
Those forms of reference are needed to represent cos
Ψ
l
and cos
Ψ
r
, the cosine of the angles between
the tangent vector
C
1
and
c
1
, respectively, and the tangent vector
D
1
and
d
1
, respectively
(also called “Cartan 1” and “Darboux 1”) by means of the scalar product
X
|
x
|
C
1
and
c
1
,
respectivel
y. S
econd, according to Bo
x
1
.54
,wederiveth
eb
asicrelationscos
Ψ
l
=
√
G
11
U
and
sin
Ψ
l
=
√
G
22
V
as well as cos
Ψ
r
=
√
g
11
u
and sin
Ψ
r
=
√
g
22
v
.
U
,V
}
u
,v
}
express the
derivative of the parameterized curve
C
(
S
)and
c
(
s
), respectively, with respect to the canonical
curve parameters
{
and
{
. Third, outlined in Box
1.55
, by means of the chain
rule, we succeed to derive
{U
,V
}
and
{u
,v
}
, respectively, in terms of the elements of the
Jacobi matrices [
∂
{
arc length
S
, arc length
s
}
] and the stretches d
s/
dS and d
S/
ds,
respectively. In this way, we succeed to represent cos
Ψ
l
and sin
Ψ
l
and cos
Ψ
r
and sin
Ψ
r
in terms
of the elements of the left and the right Cauchy-Green matrix C
l
and C
r
, respectively. Fourth,
{
U, V
}
/∂
{
u, v
}
]and[
∂
{
u,v
}
/∂
{
U, V
}
Box
1.56
leads us to the left and the right angular shear,
l
and
r
, respectively. Our great
results are presented in Corollary
1.21
. The proof follows the lines of Box
1.56
,namelythe
addition
theorem
tan(
x−y
) = (tan
x
+tan
y
)
/
(1+ tan
x
tan
y
). tan
l
(
Ψ
l
)aswellastan
r
(
Ψ
r
) establish
the optimization criteria for
maximal angular distortion
. Fifth, the characteristic optimization
problem
l
(
ψ
l
) = extr. or
r
(
ψ
r
) = extr. is dealt with in Box
1.57
. Indeed, we find the two
stationary points
tan
Ψ
l
md tan
Ψ
r
. These stationary solutions lead us to the extremal values of
l
and
r
. The celebrated representations
sin
l
sin
r
=
Λ
1
Λ
2
Λ
1
+
Λ
2
−
λ
1
λ
2
λ
1
+
λ
2
−
=
±
versus
±
(1.324)
From these extremal values of left and right angular shear
l
and
r
, we derive the left and
right
maximal angular distortion Ω
l
and
Ω
r
, respectively, namely
Ω
l
=2arcsin
versus
Ω
r
=2arcsin
Λ
1
−
Λ
2
Λ
1
+
Λ
2
λ
1
−
λ
2
λ
1
+
λ
2
(1.325)
based upon the symmetry
l
=
−
r
and
Ω
l
:=
l
−
l
,
Ω
r
:=
r
−
r
.
Indeed,
Ω
l
and
Ω
r
are the maximal data of angular distortion.
−
l
,
r
=
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