Coordinates - Map Projections: Cartographic Information Systems - page 140

Coordinate transformations:

x ( u,v )= f ( v ) e 1 ∗ (cos Ω cos u +sin Ω sin u )
+ f ( v ) e 2 ∗ ( − sin Ω cos u +cos Ω sin u )+ e 3 ∗ g ( v ) ,

(3.24)

x ( u,v )= f ( v ) e 1 ∗ cos( u − Ω )+ f ( v ) e 2 ∗ sin( u − Ω )+ e 3 ∗ g ( v ) ,
v = v ∗ , x ( u,v )= x ( u ∗ ,v ∗ )
⇔

cos u ∗ =cos( u Ω ) , sin u ∗ = sin( u Ω ) , tan u ∗ =tan( u

− − − Ω ) (3.25)

⇔
u ∗ = u − Ω.

Arc length (first differential invariant):
d s 2 =[d u, d v ]J x J x d u

,

d v

⎡ ⎤ ⎡ ⎤

⎦ , G:=J x J x = f 2

0 f 2 + g 2 .

f sin uf cos u
f cos uf sin u
0

D u xD v x
D u yD v y
D u zD v z

−

0

⎣ ⎦ = ⎣

J x = (3.26)

g

1st version: 2nd version:

d s ∗ 2 = f ∗ 2 d u ∗ 2 + f ∗ 2 + g ∗ 2 d v ∗ 2 . d s 2 = f 2 d u 2 + f 2 + g 2 d v 2 .

d u ∗ 2 =d u 2 , d v ∗ 2 =d u 2 ,

u ∗ = u Ω,v ∗ = v

− ⇔ (3.27)

d s 2 = f 2 d u 2 + f 2 + g 2 d v 2 = f 2 d v ∗ 2 +( f 2 + g 2 )d v ∗ 2 =d s ∗ 2 .

Killing vector of symmetry (rotation axis):

⎡ ⎤ ⎡ ⎤

0
0
1

0
0
1

⎣ ⎦ ∼ ⎣ ⎦ .

e 3 =[ e 1 , e 2 , e 3 ] (3.28)

Question 2: “Let a transformation group act on the coor-
dinate representation of a sphere. Or we may say, we make
a coordinate transformation. What are the transformation
groups (the coordinate transformations) which leave the
first differential invariant d s 2 of a sphere equivariant or
form-invariant ?” Answer 2: “The transformation group,
which leaves the first differential invariant d s 2 (also called
“arc length”) equivariant is the three-dimensional rotation
group R( α,β,γ ), a subsequent rotation around the 1 axis,
the 2 axis, and the 3 axis of the ambient space

3 ,δ ij }

.The
three axes establish the three Killing vectors of symmetry.”

{ R