From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 37

Examples for the mapping between two Riemann manifolds are the following. Gauss ( 1822 , 1844 )
presented his celebrated conformal mapping of the biaxial ellipsoid

2
A 1 ,A 1 ,A 2

2

E = M

l onto the

2 2

sphere S

r =

M

r , also called double projection due to a second conformal mapping of the sphere

2 2 . Amalvict and Livieratos ( 1988 ) elaborated the isoparametric mapping of

S

r onto the plane

R

2
A 1 ,A 2 ,A 3

2 2

A 1 ,A 1 ,A 2

2

the triaxial ellipsoid

r . Dermanis et al.
( 1984 ) mapped the geoid onto the biaxial ellipsoid. While nearly all existing map projections are
analyzed by means of the Cauchy-Green deformation tensor, Dermanis and Livieratos ( 1993 )used
the Euler-Lagrange deformation tensor for map projections, in particular, dilatation tr [E l G − l ]
or tr [E r G − r ] and general shear (tr[E l G − l ]) 2

E = M

l onto the biaxial ellipsoid

E = M

4det[E l G − l ] or (tr[E r G − r ]) 2 4det[E r G − r ]. An

elaborate example is discussed in Sect. 1-5 . However, to give you some breathing time, please first
enjoy the Berghaus star projection presented in Fig. 1.16 .

− −

Fig. 1.15. Right Euler-Lagrange tensor, κ 1 > 0 ,κ 2 > 0, right Euler-Lagrange circle

S 1 , right Euler-Lagrange

hyperbola H 1 √ κ 1 , √ κ 2 , left and right focal points F l and F r

Fig. 1.16. Berghaus star projection, shorelines of a spherical Earth, 18 ◦ graticule, central meridian 90 ◦ W,
“world map”