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isometric embedding
E
A
1
,A
2
⊂{
R
3
,δ
ij
}
of the
ellipsoid of revolution
in the ambient
Euclidean
space {
R
3
,δ
ij
}
is parameterized through Eqs.(
22.13
)-(
22.16
)by
Gauss surface normal coordi-
nates (L, B)
of type longitude
L
and latitude
B
. The ambient Euclidean space
{
R
3
,δ
ij
}
includes
the metric of the
left manifold
M
l
, by means of the arc length squared. Equation (
22.17
)and
the coordinates of the
left metric tensor
G
l
,Eqs.(
22.18
)-(
22.20
), in terms of the principal cur-
vature functions
M
(
B
)and
N
(
B
), respectively. Finally Eq.(
22.21
) is a representation of the arc
length squared of
A
1
,A
2
in terms of
Gauss surface normal coordinates (L, B)
. In contrast, by
means of Eq.(
22.22
) we define the algebraic geometry of the plane
E
P
2
parameterized by
Cartesian
coordinates
(
x, y
)
r
is given by the arc length squared
Eqs.(
22.24
)-(
22.26
), in terms of
Cartesian coordinates,
namely one or zero, respectively. The
left
Jacobi map
J
l
,Eq.(
22.27
), parameterizes the pullback operation (
dL, dB
)
∈
R
2
. The metric of the
right manifold
M
(
dx, dy
) leading
to the
arc length squared representation dS
2
,Eqs.(
22.20
)and(
22.29
), in terms of
Gauss surface
normal coordinates (L, B)
if the mapping equations
x
(
L, B
)
,y
(
L, B
) are given. The pullback of
(
L, B
)
to
(
x, y
) leads to the right metric tensor
C
l
,Eqs.(
22.30
)-(
22.32
), called
left Cauchy-Green
deformation tensor
C
l
. Indeed by means of Eq.(
22.33
), in terms of
left coordinates
(
L, B
).
→
M
1
:
E
2
A
1
,A
2
“
topology of
E
2
A
1
,A
2
(
left manifold
M
1
)”
Table 22.1
Metric topology of the left manifold
E
2
A
1
,A
2
:=
X
X
2
+
Y
2
A
1
∈
R
3
|
+
Z
2
/A
2
=1
,A
1
∈
R
+
,
R
+
A
2
<A
1
(22.12)
X
=
E
1
X
(
L,B
)+
E
2
Y
(
L,B
)+
E
3
Z
(
L,B
)
(22.13)
A
1
1
X
(
L, B
)=
cos
B
cos
L
(22.14)
E
2
sin
2
B
−
A
1
1
Y
(
L, B
)=
cosBsinL
(22.15)
E
2
sin
2
B
−
E
2
)
A
1
(1
−
1
Z
(
L, B
)=
sinB
(22.16)
E
2
sin
2
B
−
E
2
A
1
,A
2
”
“
left metric of
dS
2
=
G
11
dL
2
+
G
22
dB
2
(22.17)
G
11
:=
∂X
=
∂X
∂L
A
1
cos
2
B
E
2
sin
2
B
=
N
2
cos
2
B
∂L
|
(22.18)
1
−
G
22
:=
∂X
=
A
1
(1
E
2
)
2
∂X
∂B
−
=
M
2
∂B
|
(22.19)
E
2
sin
2
B
)
3
(1
−
G
12
:=
∂X
= 0
∂X
∂L
∂B
|
(22.20)
dS
2
=
N
2
cos
2
BdL
2
+
M
2
dB
2
(22.21)
By means of Figure
22.1
we want to illustrate the contents of Theorem
22.1
.The
objec-
tive function,
given by Eq.(
22.2
), to be minimized has been chosen as the distortion energy
over a
Reference Meridian Strip
with respect to the
International Reference ellipsoid
E
A
1
,A
2
by Eqs. (
22.3
)and(
22.4
). The
Meridian of Reference
has been fixed to the
Reference Longitude
L
0
=(
L
l
+
L
r
)
/
2. It turns out that the energy functional is dependent on the
l
2
-norms
Grad x
2
2
. According to the
Gauss notation / Gauss conversion /
all quantities relating to
the
left manifold
and
Grad y
l
, namely the ellipsoid of revolution, have been written in
capital letters,
while
M
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