Geography Reference
In-Depth Information
M
2
of type church tower of onion shape
Z
2
,
vertical section
,
Fig. 23.26.
Two-dimensional Riemann manifold
of radius
g
(
V
) (normal placement)
C
g
(
V
)
projective design of the
circular cylinder
map projections
in cases where the structure of the map (
23.99
)isgiven:The
first generic step
is based upon the “
left Jacobi map
” being represented by the matrix (
23.100
)ofBox
23.24
,
namely the
first derivatives
subject to the
condition
(4.6ii) which preserves the orientation of
the differential map (
dx, dy
)
(
dU, dV
) also called “
pullback
”. The representation of the “
left
Cauchy-Green deformation tensor
”
C
l
:=
J
l
G
r
J
l
subject to the “
right metric tensor
”
G
r
=
I
2
(unit matrix) of the
developed
circular cylinder
C
g
(
V
)
is the target of the
second generic step
outlined by (
23.103
). In contrast, for the
third generic step
we compare the “
left metric tensor
”
G
l
of the church tower of the onion shape, namely (
23.106
), with the “
left Cauchy-Green deformation
tensor
”
C
l
,namely(
23.102
), by means of a simultaneous diagonalization of the
pair
of positive
definite matrices
{C
l
,G
l
}
which leads to the
general eigenvalue problem
(
23.104
) with respect to
the matrices
C
l
given by (
23.102
)and
G
l
given by (
23.106
) subject to (23.4.10). As a result the
eigenvalues {Λ
1
Λ
2
}
of type (
23.110
) are computed by solving the
characteristic equation
(
23.109
).
The
fourth step
leads us to the postulates of
(
→
) a conformal mapping (conformeomorphism): Box
23.25
Λ
1
=
Λ
2
,
(
β
) an equiareal mapping (areomorphism): Box
23.26
.
Λ
1
Λ
2
=1
(
α
γ
) an equidistant mapping: Box
23.27
Λ
2
=1
.
For the
case
(
)ofa
conformal mapping
the “
canonical postulate
”
Λ
1
=
Λ
2
, the identity of the
eigenvalues (left principal stretches) leads us to (
23.111
)asthe
first order differential equation,
whichissolvedby(
23.112
). The integration constant
c
is fixed by the
boundary condition
(
23.113
),
α
namely
y
V
=
f
V
=0for
V
=
arccos
n/
(
n
+ 1). The final
mapping equation y
=
f
(
V
)
,
namely (
23.114
) is left as an integral. Indeed a closed-form integral solution exists and has
Search WWH ::
Custom Search