Geography Reference
In-Depth Information
k
r
=
1
r
2
= constant
,
Δ
ln
λ
r
+
2
r
2
λ
r
=0
, λ
r
=cosh
−
2
(
q/r
)
,
ln
λ
r
=
−
2lncosh(
q/r
)
,
(1.220)
2
2
r
2
1
cosh
2
(
q/r
)
D
q
ln
λ
r
=
r
tanh(
q/r
))
, Δ
r
ln
λ
r
=
D
qq
ln
λ
r
=
−
−
2
r
2
λ
r
q. e. d.
Left differential equation of the factor of conformality (
=
−
2
A
1
,A
1
,A
2
E
):
k
l
=
(1
−
E
2
sin
2
φ
)
2
=
1
−
E
2
sin
2
f
−
1
(
Q/A
1
)
,
A
1
(1
A
1
(1
−
E
2
)
−
E
2
)
cos
2
f
−
1
(
Q/A
1
)
Δ
ln
λ
l
+2
k
(
Q
)
λ
l
=0
, λ
l
=
1
− E
2
sin
2
f
−
1
(
Q/A
1
)
,
(1.221)
1
ln
λ
l
=2ln
f
−
1
(
Q/A
1
)
−
2
ln[1
− E
2
sin
2
f
−
1
(
Q/A
1
)]
,
Δ
l
ln
λ
l
=
D
QQ
ln
λ
l
=
2
k
(
Q
)
λ
l
−
q.e.d
.
In Box
1.27
, we write down the metric forms “left d
S
2
” and “right d
s
2
” in the initial coordinates
{
E
2
sin
2
Φ
) and (ii) cos
2
φ
.
The first term
A
1
d
Λ
and
r
d
λ
, respectively, generates d
P
and d
p
, respectively. In contrast, the
second term ([
A
1
(1
, respectively. Second, we factorize by (i) cos
2
Φ/
(1
Λ, Φ
}
and
{
λ, φ
}
−
E
2
sin
2
Φ
)] cos
Φ
)d
Φ
and (
r/
cos
φ
)d
φ
, respectively, generates d
Q
and d
q
, respectively. Indeed, the first factors cos
2
Φ/
(1
E
2
)
/
(1
−
−
E
2
sin
2
Φ
)andcos
2
φ
produce the left
and the right factor of conformality, called
λ
l
and
λ
r
, respectively. They are reciprocal to
Λ
l
and
Λ
r
, respectively. Third, by means of Box
1.28
, we aim at representing the factors of conformality,
λ
l
and
λ
r
, in terms of conformal (isometric, isothermal) latitude
Q
and
q
, respectively, namely
λ
l
(
Q
)and
λ
r
(
q
), respectively. Here, we have to invert the functions
Q/A
1
=
f
(
Φ
)and
q/r
=
ln tan(
π/
4+
φ/
2) = artanh (sin
φ
), also called the inverse
Lambert
or
Gudermann function
, lam or
gd, respectively,
φ
=lam(
q/r
)=gd(
q/r
)orsin
φ
= tanh(q/r), cos
φ
=1
/
cosh(q
/
r). While
λ
l
(
Q
)
and
Λ
l
(
Q
) cannot be given in a closed form,
λ
r
=1
/
cosh
2
(
q/r
)and
Λ
r
=cosh
2
(
q/r
) are available
in a simple form. Fourth, by means of Box
1.29
, we prove that
λ
r
and
λ
l
, respectively, fulfill the
conformal representation of the right and the left Gaussian curvature, here written in two versions
as a special Helmholtz differential equation. For being simpler, we did first “right” followed by
the more complex “left” computation. Indeed, for given Gaussian curvature
k
r
=1
/r
2
= constant
of the sphere
−
2
E
2
sin
2
Φ
)
2
/
[
A
1
(1
E
2
)] of the ellipsoid-of-revolution
2
A
1
,A
1
,A
2
S
r
and
k
l
=(1
−
−
E
finally transformed into
coordinates of type conformal (isometric, isothermal), we succeed
to prove
Δ
ln
λ
2
+2
kλ
2
= 0 of type “right” and “left”.
{
q,Q
}
Solution (the fourth problem).
2
2
A “simple conformal mapping” of
E
A
1
,A
1
,A
2
→
S
r
is the isoparametric mapping characterized by
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