Geography Reference
In-Depth Information
Fig. I.1.
Airy optimal dilatation factor
ρ
for a symmetric strip [
Λ
W
,Λ
E
]
−Φ
N
,Φ
N
] with
Λ
W
=
Λ
0
−ΔΛ
and
Λ
E
=
Λ
0
+
ΔΛ
, generalized UM, WGS 84,
ρ
(
Φ
N
): Airy optimal dilatation factor as function of the half-strip
width
Φ
N
×
[
Φ
S
=
R
=
ρ
0
A
1
cos
Φ
0
/
1
− E
2
sin
2
Φ
0
,
(I.31)
A
1
,A
2
:=
X
∈
R
3
+
,A
1
>A
2
,
X
2
+
Y
2
A
1
+
Z
2
2
+
,A
2
∈
R
X
∈
E
A
2
=1
,A
1
∈
R
(I.32)
x
=
ρ
0
A
1
cos
Φ
0
(
Λ
−
Λ
0
)
1
,
E
2
sin
2
Φ
0
−
(I.33)
ln
tan
π
E/
2
.
1
cos
Φ
0
4
+
Φ
E
sin
Φ
1+sin
Φ
−
y
=
ρ
0
A
1
1
2
E
2
sin
2
Φ
0
−
The plane covered by the chart (
x, y
). Cartesian coordinates, with an Euclidean metric, namely
{
R
2
,δ
kl
}
2
(Kronecker delta, unit matrix) is generated by developing the circular cylinder
C
R
of
radius (
I.31
) with respect to the surface normal latitude
Φ
0
of reference.
End of Definition.
Lemma I.9 (Generalized polycylindric projection, principal stretches).
With respect to the left Tissot distortion measure represented by the matrix C
l
G
−
l
of the left
Cauchy-Green deformation tensor C
l
=J
l
G
r
J
l
multiplied by the inverse of the left metric ten-
sor G
l
, the matrix of the metric tensor of
2
A
1
,A
2
E
, the left principal stretches of the generalized
polycylindric projection are given by
1
E
2
sin
2
Φ
cos
Φ
cos
Φ
0
−
1
Λ
1
=
Λ
2
=
ρ
0
.
(I.34)
E
2
sin
2
Φ
0
−
cover the eigenspace of the left Tissot matrix C
l
G
−
l
. Due to conformality,
they are identical,
Λ
1
=
Λ
2
=
Λ
S
,
J
l
denotes the left Jacobi map (d
x,
d
y
)
The eigenvalues
{
Λ
1
,Λ
2
}
→
(d
Λ,
d
Φ
)
,
G
r
the
Search WWH ::
Custom Search