Geography Reference
In-Depth Information
Up to now, all deformation measures have been built on the
first differential invariants I
l
and
I
r
of
surface geometry
, which are also called d
S
2
and d
s
2
. Such an invariant “left” or “right”
measures the infinitesimal distance between two points on the “left” or the “right” surface. A
dual measure of a surface (two-dimensional Riemann manifold) immersed in
3
is the
infinitesimal
surface element
. Indeed, the surface element “left versus right”, (
1.264
), is
dual
to the infinitesimal
distance element “left versus right”, (
1.265
):
d
S
l
:=
det[G
l
]d
U
R
d
V
versus d
S
r
:=
det[G
r
]d
u
∧
∧
d
v,
(1.264)
d
S
2
=
G
11
d
U
2
+2
G
12
d
U
d
V
+
G
22
d
V
2
versus d
s
2
=
g
11
d
u
2
+2
g
12
d
u
d
v
+
g
22
d
v
2
.
(1.265)
2
l
2
In the context of the mapping
f
:
M
→
M
r
, we next define
areomorphism
as an
equiareal mapping
2
l
2
M
→
M
r
: see Definition
1.13
.
Definition 1.13 (Equiareal mapping).
l
r
is called
area preserving
and
equiareal
An orientation preserving diffeomorphism
f
:
M
→
M
(vector product preserving,
areomorphism
)if
det[G
l
]d
U
d
V
=
det[G
r
]d
u
∧
∧
d
v
(1.266)
or, equivalently,
Φ
l
=
Φ
2
:=
det[G
r
]d
u
∧
d
v
det[G
l
]d
U
d
V
=1
∧
⇔
(1.267)
1=
det[G
l
]d
U
∧
d
V
=:
Φ
2
=
Φ
r
det[G
r
]d
u
∧
d
v
or
S
l
r
:=
det[G
r
]d
uΛ
d
v
det[G
l
]d
U
−
∧
d
V
=0
⇔
S
r
l
:=
det[G
l
]d
U
det[G
r
]du
∧
d
V
−
∧
d
v
= 0
(1.268)
2
2
for all points of
M
l
and
M
r
.
respectively, holds.
End of Definition.
Indeed, the left surface element
det[G
l
]d
U
d
V
as well as the right surface element
det[G
r
]d
u
∧
d
v
have enabled us to introduce
dual measures
to the left length element
dU
T
G
l
d
U
as well as
to the right length element
du
T
G
r
d
u
There exist representations of the multiplicative measure of
areal distortion,
∧
Φ
l
,Φ
r
}
,intermsof
the Cauchy-Green deformation tensor, the Euler-Lagrange deformation tensor, and the principal
stretches (left or right eigenvalues), which we collect in Box
1.39
and turn out to be useful in the
equivalence theorem.
{
and of the additive measure of areal distortion,
{
S
l
r
,S
r
l
}
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