Geodesics - Map Projections: Cartographic Information Systems - page 722

U M + G KL U K U L U M +3 M

U K U L +2 G KL L

PQ

U K U L U P U M +

KL

+ M

U K U L U P + G KL

+ Q

M
QP

K
PQ

L
ST

U P U Q U S U T U M =0 .

KL KL

,P

Compare our result with Vogel ( 1970 , p . 642 , formula 3) .

End of Proof.

E-3 The Newton Form of a Geodesic in Maupertuis Gauge (Sphere,
Ellipsoid-of-Revolution)

Geodesics , in particular minimal geodesics , are of focal geodetic interest. In terms of Riemann
normal coordinates, they are used in map projections to establish azimuthal maps —maps on
a local tangential plane T p M

—which are geodetic with respect to the point p of evaluation.
Straight lines in a Riemann map (plane chart) are the shortest on the surface, for example the
Earth, with respect to the point p of evaluation. In geodetic navigation—aerial navigation, space
navigation—minimal geodesies are applied to connect points on the Earth surface or in space. In
both applications, initial value problems, boundary value problems as well as their mixed forms
play the dominant role in solving the differential equations of a geodesic. (See Lichtenegger ( 1987 )
for a review of the four fundamental geodesic problems.)

Section E-31.

The differential equations of a geodesic can be written either as a system of two differential
equations of second order (Lagrange portrait) or as a system of four differential equations of first
order (Hamilton portrait) as long as we refer to two-dimensional surfaces. The Lagrange portrait
and the Hamilton portrait of a geodesic is presented in the first section.

Section E-32.

Recently, Goenner et al. ( 1994 ) have shown that the Newton law balancing inertial forces, in
particular accelerations, and acting forces, in particular, those forces which are being derived
from a potential, can be interpreted as a set of three geodesics in a three-dimensional Riemann
manifold. In this case, the three-dimensional Riemann manifold is parameterized by conformal
coordinates (isometric coordinates), in short, the three-dimensional Riemann manifold is said to
be conformally flat . In addition, the factor of conformality represents as a Maupertuis gauge the
potential of conservative forces. In turn, we try in the second section to express geodesics firstly by
conformal coordinates (isometric coordinates) and secondly by Maupertuis gauge, in particular,
aiming at a representation of geodesics in their Newton form (Newton portrait).