Geography Reference
In-Depth Information
Fig. 20.1.
Gnomonic projection of the sphere, the
straight lines
are geodesics (great circles), the loxodromes
(rhumblines) are
circular
. Great circle (1) and rhumb line (2)
Section 20-1.
The elaborate presentation of
Riemann polar/normal coordinates
starts in Sect.
20-1
by the setup
of a
minimal atlas
of the biaxial ellipsoid, namely in terms of
{
ellipsoidal longitude, ellipsoidal
latitude
}
and
{
meta-longitude, meta-latitude
}
.Boxes
20.1
and
20.2
collect all fundamental ele-
(two-dimensional ellipsoid-of-revolution, semi-major axis
A
1
,
semi-minor axis
A
2
). The
Darboux frame
of a one-dimensional submanifold in the two-dimensional
manifold
E
ments of surface geometry of
E
A
1
,A
2
A
1
,A
2
is reviewed, in particular, by Corollary
20.3
, the representation of geodetic cur-
vature, geodetic torsion, and normal curvature in terms of elements of the first and second funda-
mental form as well as of Christoffel symbols. First, we define the
geodesic
. Second, we define the
geodesic circle
following
Fialkow
(
1939
),
Schouten
(
1954
),
Vogel
(
1970
,
1973
)and
Yano
(
1940a
,
1942
) enriched by two examples. Corollary
20.2
states that a curve is a
geodesic
if and only if it
fulfills a system of
second order
ordinary differential equations (
20.42
). In contrast, a curve is a
geodesic circle
if and only if it fulfills a system of
third order
ordinary differential equations (
20.43
).
Proofs are presented in Appendix
E-1
and
E-2
. Finally, we define
Riemann polar/normal coordi-
nates
and by Definition
20.4
the Riemann mapping.
Section 20-2.
Section
20-2
concentrates on the computation of Riemann polar/normal coordinates. First, by
solving the two
second order
ordinary differential equations of a geodesic in the
Lagrange por-
trait
, namely by means of the
Legendre recurrence
(“Legendre series”), in particular, initial value
problem versus boundary value problem, by the technique of
standard series inversion
. Second,
the three
first order
ordinary differential equations of a geodesic in the
Hamilton portrait
(“phase
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