Coordinates - Map Projections: Cartographic Information Systems - page 142

e 2 = e 1 ∗ (sin γ cos β )+ e 2 ∗ (cos γ cos α +sin γ sin β sin α )+
+ e 3 ∗ ( − cos γ sin α +sin γ sin β cos α ) ,

e 3 = e 1 ∗ ( − sin β )+ e 2 ∗ (cos β sin α )+ e 3 ∗ (cos β cos α ) .

Coordinate transformations:

x ( u,v )= x ( u ∗ ,v ∗ )
⇔

e 1 ∗ f 1 ( α,β,γ | u,v )+ e 2 ∗ f 2 ( α,β,γ | u,v )+ e 3 ∗ f 3 ( α,β,γ | u,v ) = (3.32)

= e 1 ∗ cos v ∗ cos u ∗ + e 2 ∗ cos v ∗ sin u ∗ + e 3 ∗ sin v ∗ ,
cos v ∗ cos u ∗ = f 1 ( α,β,γ|u,v )=
=cos γ cos β cos v cos u +sin γ cos β cos v sin u − sin β sin v,
cos v ∗ sin u ∗ = f 2 ( α,β,γ|u,v )=
= − (sin γ cos α +cos γ sin β sin α )cos v cos u +
+(cos γ cos α +sin γ sin β sin α )cos v sin u +cos β sin α sin v, (3.33)

sin v ∗ = f 3 ( α,β,γ

u, v )=
=(sin γ sin α +cos γ sin β cos α )cos v cos u

|

−

− (cos γ sin α +sin γ sin β cos α )cos v sin u +cos β cos α sin v,

tan u ∗ = f 2 /f 1 , sin v ∗ = f 3 . (3.34)

Arc length (first differential invariant):

d s 2 =[d u, d v ] r 2 cos v 0

r 2 d u

(“diffeomorphism”) , d u ∗

=J d u ,

d v ∗

0 d v d v

J:= D u u ∗ D v u ∗

,

D u v ∗ D v v ∗

dtan u ∗ =(1+tan 2 u ∗ )d u ∗ ⇒ d u ∗ =cos 2 u ∗ dtan u ∗ ,

1
1 − sin 2 v ∗ dsin v ∗ =cos v ∗ d v ∗ ⇒ d v ∗ = dsin v ∗ , (3.35)

d s 2 = r 2 cos 2 v d u 2 + r 2 d v 2 = r 2 cos 2 v ∗ d v ∗ 2 + r 2 d v ∗ 2 =d s ∗ 2 .

Killing vector of symmetry (rotation axis):

⎡ ⎤ ⎡ ⎤

1
0
0

0
1
0

⎣ ⎦ , 2 axis of symmetry : e 2 ∼ ⎣ ⎦ ,

1 axis of symmetry : e 1 ∼

⎡ ⎤

0
0
1

⎣ ⎦ .

3 axis of symmetry : e 3 ∼ (3.36)