Geodesics - Map Projections: Cartographic Information Systems - page 723

Section E-33, Section E-34.

The program to express the minimal geodesies as the Newton law is realized for geodesics in
the sphere

2 R in the third section and in the ellipsoid-of-revolution A 1 ,A 2

S E in the fourth section.

Extensive numerical examples in the form of computer graphics are given.

Section E-35, Section E-36.

We refer to Sect. E-35 for a review of Maupertuis gauged geodesics parameterized in normal
coordinates with respect to a local tangent plane. We refer to Sect. E-36 for a review of Lie series,
Maupertuis gauged geodesics, and the Hamilton portrait.

E-31 The Lagrange Portrait and the Hamilton Portrait of a Geodesic

2 ,g μν

Let there be given a two-dimensional Riemann manifold { M } with the standard metric

2 × 2 , symmetric and positive-definite, shortly a surface . For geodetic
applications, we shall assume the following important properties.

GgivenbyG= { g μν }∈ R

{ M

2 ,g μν } : orientable, star-shaped, second order Holder
continuous, compact.

2 ,g μν

Thus, we have excluded corners, edges, and self-intersections. is totally covered by

a set of charts which form by their union a complete atlas. Any chart is an open subset of a
two-dimensional Euclidean manifold

{ M }

2 := 2 ,δ μν

,
the unit matrix. Indeed, we shall deal only with topographic surfaces which are assumed to be
topologically similar to the sphere. Thus, a minimal atlas of

E { R } with standard canonical metric I = { δ μν }

2 ,g μν

is established by two charts,
for example, as described by Engels and Grafarend ( 1995 ), based upon the following coordinates.

{ M }

Quasi-spherical coordinates. Meta-quasi-spherical coordi-
nates.

According to a standard theorem, for example, Chern ( 1955a ) applied to two-dimensional Rie-
mann manifolds, conformal (isothermal, isometric) coordinates {q 1 ,q 2 } always exist. They estab-

lish a conformal diffeomorphism { M

2 ,g μν }→{ R 2 ,δ μν } which is angle preserving. In the fol-

lowing, we shall adopt to be conformally flat , in particular g μν = λ 2 ( q 1 ,q 2 ) δ μν ,where

λ 2 ( q 1 ,q 2 ) is the factor of conformality. The infinitesimal distance between two points ( q,q +d q )
in

{ M

2 ,g μν }

can correspondingly be represented by ( E.17 )and( E.18 ). Obviously, d q μ is an
element of the tangent space T q M

{ M

2 ,λ 2 δ μν }

2 at q , while g μν d q ν is an element of the cotangent space ∗ T q M 2 ,

2 := 2 , G q }

which is the dual space of T q M { R .