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M
2
=
T
2
A,B
;
A
:
B
=4:1
,α
=
β
=
γ
=20
◦
Fig. 23.1.
Oblique orthogonal projection of the torus
M
2
of type torus
T
2
A,B
,
vertical section
, projective design of the
Fig. 23.2.
Two-dimensional Riemann manifold
T
π/
2
M
2
at
V
=
C
2
A
+
B
of radius
central plane
(parallel to the tangent plane
±
π/
2andofthe
circular cylinder
A
+
B
(normal placement)
We begin with a general set-up of the mapping equations for
case
(
i
) central plane and
case
(
ii
) circular cylinder by
x
y
=
f
(
V
)
cos U
or α
=
U, r
=
f
(
V
)
(23.4)
sin U
x
y
=
(
A
+
B
)
U
(23.5)
g
(
V
)
2
.Thus
we are left with the problem to
determine the unknown functions f(V)
for
case
(
i
)aswellas
g(V) for
case
(
ii
). Here we follow the
constructive approach
of the
theory of map projections
in cases where the structure of the map (
23.4
)aswellas(
23.5
) is given. Its generic steps are
outlined in Box
23.2
:The
first generic step
isbaseduponthe“
left Jacobi map
” being represented
by the matrices (
23.6b
)and(
23.6c
)of
first derivatives
subject to the
condition
(
23.6d
)which
preserves the orientation of the differential map (
dx, dy
)
either in terms of
Cartesian coordinates
(
x, y
)or
polar coordinates
(
α,r
)whichcover
R
(
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
case
(
i
) central plane or
case
(
ii
)the
developed
circular
cylinder is the target of the
second generic step.
In contrast, for the
third generic step
we compare
the “
left metric tensor
”
G
l
of the torus
→
2
A,B
with the “
left Cauchy-Green deformation tensor
”
C
l
, namely by a simultaneous diagonalization of the
pair
of positive define matrices
T
}
which leads to the
general eigenvalue problem
(
23.8a
) with respect to the matrices
C
l
given
{
C
l
,G
l
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