Geography Reference
In-Depth Information
r
Λ, q
=
Q, A
1
ln
tan
π
1
E/
2
p
=
P, λ
=
A
1
4
+
Φ
E
sin
Φ
1+
E
sin
Φ
−
2
=
r
ln tan
π
.
4
+
φ
2
(1.222)
Gauss
(
1822
,
1844
) made some special proposals how to choose the radius
r
of
S
r
in an optimal
way. Here, let us refer to Chap.
2
, where the
Gauss projection
E
2
is discussed in
detail. Here, we conclude with a representation of the left as well as the right inverse mapping
Φ
−
1
A
1
,A
1
,A
2
→
S
r
→
P
l
and
Φ
−
r
in terms of conformal coordinates (isometric, isothermal) of Box
1.30
, which specializes
Φ
−
l
and
Φ
−
r
of Box
1.21
.
End of Solution (the fourth problem).
Box 1.30 (Representation of
Φ
−
l
and
Φ
−
r
in terms of conformal coordinates:
2
2
E
A
1
,A
1
,A
2
→
S
r
).
Φ
−
l
:
X
(
Λ, Φ
)=
Φ
−
r
:
x
(
λ, φ
)=
=
E
1
A
1
cos
Φ
cos
Λ
1
− E
2
sin
2
Φ
+
=
e
1
r
cos
φ
cos
λ
+
+
E
2
A
1
cos
Φ
cos
Λ
1
+
+
e
2
r
cos
φ
sin
λ
+
E
2
sin
2
Φ
−
E
2
)sin
Φ
+
E
3
A
1
(1
−
1
=
+
e
3
r
sin
φ
=
(1.223)
E
2
sin
2
Φ
−
=
E
1
A
1
cos
f
−
1
(
Q/A
1
)cos(
P/A
1
)
+=
e
1
r
cos(
p/r
)
1
cosh(
q/r
)
+
E
2
sin
2
f
−
1
(
Q/A
1
)
−
+
E
2
A
1
cos
f
−
1
(
Q/A
1
) sin(
P/A
1
)
++
e
2
r
sin(
p/r
)
1
− E
2
sin
2
f
−
1
(
Q/A
1
)
cosh(
q/r
)
+
E
2
)sin
f
−
1
(
Q/A
1
)
1
− E
2
sin
2
f
−
1
(
Q/A
1
)
+
E
3
A
1
(1
−
.
+
e
3
r
tanh(
q/r
)
.
Isoparametric mapping:
p
=
P
and
q
=
Q.
2
A
1
,A
1
,A
2
2
Example 1.12 (Conformal mapping of an ellipsoid-of-revolution
E
toasphere
S
r
:the
2
A
1
,A
1
,A
2
2
Universal Stereographic Projection (UPS) of type left
r
, special Korn-
Lichtenstein equations, Cauchy-Riemann equations (d'Alembert-Euler equations)).
E
and right
S
Let us assume that we have found a solution of the left Korn-Lichtenstein equations of the
ellipsoid-of-revolution
2
A
1
,A
1
,A
2
E
parameterized by the two coordinates
{
Λ, Φ
}
which convention-
ally are called
. Similarly, let
us depart from a solution of the right Korn-Lichtenstein equations of the sphere
{
Gauss surface normal longitude, Gauss surface normal latitude
}
r
parame-
S
terized by the two coordinates
{
λ, φ
}
which are called
{
spherical longitude, spherical latitude
}
.
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