Geography Reference
In-Depth Information
, called
left curves c
l
(
t
) (left one-dimensio
na
l submanifold) and
right curves c
r
(
t
) (right one-dimensional
submanifold). Under the mapping
f
{
U
= constant
V
}
,
{
U, V
= constant
}
as well as
{
u
= constant,
u
}
,
{
u,v
= constant
}
◦
c
l
(
t
)=
c
l
(
t
), the mapping
equidistant
→
M
3
2
l
1
l
1
r
2
r
3
c
l
(
t
)
→
c
r
(
t
)or
R
⊃
M
⊃
M
⊂
M
⊂
R
(1.296)
is called
equidistant
if a finite section of a specific left curve
c
l
(
t
) has the same length as a finite
section of a corresponding right curve
c
r
(
t
).
End of Definition.
Let us work out the equivalence theorem for an equidistant mapping from a left curve
c
l
(
t
)toa
right curve
c
r
(
t
).
3
2
l
1
l
1
r
2
r
3
).
Theorem 1.19 (Equidistant mapping
R
⊃
M
⊃
M
→
M
⊂
M
⊂
R
Let us assume that the left surface (left two-dimensional Riemann manifold) as well as the right
surface (right two-dimensional Riemann manifold) has been parameterized by left coordinates
{
. If the directions of their left tangent vectors and their right
tangent vectors coincide with the directions of the left principal stretches (left eigendirections,
left eigenvectors) and of the right principal stretches (right eigendirections, right eigenvectors),
then the following conditions of an equidistant mapping are equivalent.
(i) Equidistant mapping of a section of a specific left curve
c
l
(
t
) to a corresponding section of a
specific right curve
c
r
(
t
).
U, V
}
and right coordinates
{
u,v
}
U
coord
inate l
ine to
u
coor
dinate
line :
V
coor
dinate
line to
v
coor
dinate
line:
G
22
(
t
)
V
d
t
=
b
r
g
22
(
t
)
v
d
t.
G
22
(
t
)
U
d
t
=
b
r
g
11
(
t
)
u
d
t.
b
l
a
l
b
l
a
l
(1.297)
a
r
a
1
(ii) Left or right Cauchy-Green matrix under an equidistant mapping
c
l
(
t
)
→
c
r
(
t
).
U
coordinate line to
u
coordinate line:
c
22
=
G
22
or
C
22
=
g
22
V
coordinate line to
v
coordinate line:
c
11
=
G
11
or
C
11
=
g
11
.
(1.298)
(iii) Left or right principal stretches under an equidistant mapping
c
l
(
t
)
→
c
r
(
t
).
U
coordinate line to
u
coordinate line:
Λ
2
=1 or
λ
2
=1
.
V
coordinate line to
v
coordinate line:
Λ
1
=1 or
λ
1
=1
.
(1.299)
End of Theorem.
The proof is straightforward. We refer to Example
1.11
, where we solved the third problem: the
ellipsoidal equator had been equidistantly mapped to the spherical equator, namely
rλ
=
A
1
Λ
,
such that
Λ
1
(
Φ
=0)=1or
λ
1
(
φ
=0)=1.
1-143 Canonical Criteria
By means of the various equivalence theorems, we are well-prepared to present to you, as beloved
collectors items of Box
1.46
, the canonical criteria or measures for a conformal, an equiareal,
and an isometric mapping
M
l
→
M
r
as well as for an equidistant mapping
c
l
(
t
)
→ c
r
(
t
). These
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