Geodetic Mapping - Map Projections: Cartographic Information Systems - page 463

20
Geodetic Mapping

Geodesics, geodetic mapping. Riemann, Soldner, and Fermi coordinates on the ellipsoid-
of-revolution, initial values, boundary values. Initial value problems versus boundary value
problems.

A global length preserving mapping of a geodetic reference surface such as the sphere or such as the
ellipsoid-of-revolution (spheroid) onto the plane (the chart) does not exist . Thus, as a compromise,
equidistant mappings of certain coordinate lines like the equator or the central meridian of a
UTM/Gauss-Krueger strip system have been proposed. Of focal interest are geodetic mappings :
a mapping of a surface (two-dimensional Riemann manifold) is called geodetic if geodesics on the
given surface (in particular, shortest geodesics like the “great circles” on the sphere) are mapped
onto straight lines in the plane (the chart). In the plane (the chart), straight lines are geodesics,
of course. According to a fundamental lemma of Beltrami ( 1866 ), a geodetic mapping of a surface
exists if and only if the surface is characterized by constant Gaussian curvature .Thus,ageodetic
mapping of the sphere does exist, for example, (the gnomonic projection ). Compare with Fig. 20.1 .

Beltrami ( 1866 ): a geodetic mapping of a surface exists if
and only if the surface is characterized by constant Gaus-
sian curvature.

Unfortunately, the ellipsoid-of-revolution (spheroid) is not of constant Gaussian curvature ;to
the contrary, its Gaussian curvature depends on ellipsoidal latitude. In this situation, Riemann
( 1851 ) has proposed to use instead a geodetic mapping with respect to one central point only:
with respect to one particular point P of contact, a tangential plane T P M

2 of the surface (two-

2 )ischosentomap P -passing geodesics equidistantly onto the

dimensional Riemann manifold M

2 . In the tangential plane T P M 2 at point P , either polar coordinates

tangential plane T P M { α, r }

or normal coordinates { x, y } = { r cos α, r sin α } are used where α is the azimuth of the geodesic

2 and r is its length .These Riemann coordinates (polar or normal) represent
length preserving mappings with respect to the central point P

passing P ∈ T P M

2 .

∈ T P M