Coordinates - Map Projections: Cartographic Information Systems - page 138

Φ 2 ( x ), the meta-North Pole (“West Pole”), the
meta-South Pole (“East Pole”), and the α = 0 meridian define the set of right exceptional points .

exceptional points . In the right chart { α,β }∈

End of Example.

S r , minimal atlas of two charts, orthogonal projections p = π 1 ( P ) ,q = π 2 ( P )

Fig. 3.5. “One-sphere”

3-2 Killing Vectors of Symmetry

Killing vectors of symmetry for the surface-of-revolution and the sphere, transformation
groups, first differential invariants, rotation group R 3 ( Ω ), special orthogonal groups SO(2)
and SO(3).

In order to understand better the special aspects of a surface, called transverse and oblique ,we
have to analyse the special symmetries of the surface-of-revolution, in particular, the ellipsoid-
of-revolution, and the sphere. Such a symmetry analysis is conventionally based on the Killing
vector of symmetry , which we are going to compute here. As soon as we have identified at least one
Killing vector of symmetry for the surface of revolution and the three Killing vectors of symmetry
of the sphere, we discuss their impact on the definition of the transverse aspect as well as of the
oblique aspect of a surface. We pose two questions.