Geography Reference
In-Depth Information
E
Geodesics
Geodetic curvature, geodetic torsion, geodesics. Lagrangean portrait, Hamiltonian por-
trait. Maupertuis gauge, Universal Lambert Projection, Universal Stereographic Projection,
dynamic time.
Three topics are presented here.
Section E-1.
In Sect.
E-1
, we review the presentation of
geodetic curvature
,
geodetic torsion
,and
normal cur-
vatures
of a submanifold on a two-dimensional Riemann manifold.
Section E-2.
In Sect.
E-2
, in some detail, we review the Darboux equations. Relatively unknown is the deriva-
tion of the differential equations of third order of a geodesic circle.
Section E-3.
In Sect.
E-3
. we concentrate on the Newton form of a geodesic in Maupertuis gauge on the sphere
and the ellipsoid-of-revolution.
E-1 Geodetic Curvature, Geodetic Torsion, and Normal Curvature
For the proof of Corollary
20.1
,wedepartfromthe
Darboux equations
(
20.11
). According
to (
20.11
), the first
Darboux vector
D
1
is represented in terms of the tangent vectors
,
thus enjoying the derivative (
E.1
). In contrast, (
E.2
) expresses the derivative of the third Darboux
vector
D
3
=
G
3
, which coincides with the
surface normal vector
.
{
G
1
,
G
2
}
Search WWH ::
Custom Search