Coordinates - Map Projections: Cartographic Information Systems - page 135

{ do not cover all points of the “two-sphere”. As a set of exceptional points, we removed

the South Pole, the North Pole, as well as the Greenwich Meridian, the meridian λ = 0. Here,
we introduce a special union of six charts, which covers the “two-sphere” completely. Figure 3.4
illustrates those six charts. Their generators Φ i = Φ (

λ, φ }

U i )( i =1 , 2 , 3 , 4 , 5 , 6) are the following:

| z =+ r 2

+ 2

r

− ( x 2 + y 2 ) > 0 ,x 2 + y 2 <r 2 U

z = { [ x, y, z ] ∈ S

},

Φ 1 ( x, y, z ):= x

= u 1 ,

y v 1

Φ − 1 ( u 1 ,v 1 )= u 1 ,v 1 , + r 2 − ( u 1 + v 1 )

∼ x 1 ( u,v )= e 1 u 1 + e 2 v 1 + e 3 r 2

− ( u 1 + v 1 ) ,

r 2

U z =

2
r

( x 2 + y 2 ) < 0 ,x 2 + y 2 <r 2 { [ x, y, z ] ∈ S | z = − − } ,

Φ 2 ( x, y, z ):= x

= u 2 ,

y v 2

Φ − 2 ( u 2 ,v 2 )= u 2 ,v 2 , ( u 2 + v 2 )

r 2

e 3 r 2 ( u 2 + v 2 ) ,

− − ∼ x 2 ( u,v )= e 1 u 2 + e 2 v 2 − −

y =+ r 2

+ 2

r

( x 2 + z 2 ) > 0 ,x 2 + z 2 <r 2

U

y =

{ [ x, y, z ] ∈ S | − } ,

Φ 3 ( x, y, z ):= x

= u 3 ,

z v 3

Φ − 3 ( u 3 ,v 3 )= u 3 , + r 2 − ( u 3 + v 3 ) ,v 3

∼ x 3 ( u, v )= e 1 u 3 + e 2 r 2

− ( u 3 + v 3 )+ e 3 v 3 ,

|y = − r 2 U y = { [ x, y, z ] ∈ S

2
r

− ( x 2 + z 2 ) < 0 ,x 2 + z 2 <r 2 }, (3.16)

Φ 4 ( x, y, z ):= x

= u 4 ,

z v 4

Φ − 4 ( u 4 ,v 4 )= u 4 , ( u 4 + v 4 ) ,v 4

r 2

e 2 r 2 ( u 4 + v 4 )+ e 3 v 4 ,

− − ∼ x 4 ( u, v )= e 1 u 4 − −

x =+ r 2

+ 2

r

( y 2 + z 2 ) ,y 2 + z 2 <r 2 U

x =

{ [ x, y, z ] ∈ S | − } ,

Φ 5 ( x, y, z ):= y

= u 5 ,

z v 5

Φ − 5 ( u 5 ,v 5 )= + r 2 ( u 5 + v 5 ) ,u 5 ,v 5

x 5 ( u,v )=+ e 1 r 2

− ∼ − ( u 5 + v 5 )+ e 2 u 5 + e 3 v 5 ,

r 2

U x =

2
r

( y 2 + z 2 ) ,y 2 + z 2 <r 2 { [ x, y, z ] ∈ S | x = − − } ,

Φ 6 ( x, y, z ):= y

= u 6 ,

z v 6

Φ − 6 ( u 6 ,v 6 )= ( u 6 + v 6 ) ,u 6 ,v 6

r 2

e 1 r 2 − − ∼ x 6 ( u,v )= − − ( u 6 + v 6 )+ e 2 u 6 + e 3 v 6 .

The sets U i and their images Φ ( U i ) are open with respect to the chosen topology. For instance,
the set U

z and its image Φ 1 ( x, y, z ):

z = [ x, y, z ]

( x 2 + y 2 ) > 0 ,x 2 + y 2 <r 2 , z =+ r 2

+ 2

r

U ∈ S | −