Coordinates - Map Projections: Cartographic Information Systems - page 141

A proof of our answer is outlined in Box 3.2 . First, we present a parameter representation of
a sphere, defined by

u ∗ ,v ∗ }

in an
oblique frame of reference generated by the three-dimensional orthogonal group SO(3). Second,
the action of the transformation group SO(3) is parameterized by Cardan angles

{ u,v } in an equatorial frame of reference and defined by {

,namely
a rotation R 1 ( α ) around the 1 axis, a rotation R 2 ( β ) around the 2 axis, and a rotation R 3 ( γ )
around the 3 axis. Third, we transform forward and backward the orthonormal system of base
vectors

{ α,β,γ }

{ e 1 , e 2 , e 3 |O} and { e 1 ∗ , e 2 ∗ , e 3 ∗ |O} , which span the three-dimensional Euclidean space,

2
r

the ambient space of the sphere S ·{ e 1 , e 2 , e 3 |O} establish the conventional equatorial frame

of reference, at the origin the meta-equatorial reference frame. Fourth, the

backward transformation is substituted into the parameter representation of the placement vector
e 1 r cos v cos u + e 2 r cos v sin u + e 3 r sin v

{ e 1 ∗ , e 2 ∗ , e 3 ∗ |O}

r , such that e 1 ∗ f 1 ( α,β,γ

∈ S | u, v )+ e 2 ∗ f 2 ( α,β,γ | u,v )+

e 3 ∗ f 3 ( α,β,γ u, v ) is a materialization of the “Kartenwechsel” (“cha-cha-cha”). In this way, we

are led to tan u ∗ = f 2 /f 1 and sin v ∗ = f 3 , both functions of the parameters

|

SO(3),
of the longitude u , and the latitude v . Fifth, as soon as we substitute “cha-cha-cha”, namely the
diffeomorphism

{ α,β,γ }∈

d u ∗ , d v ∗ }

by means of the Jacobi matrix J in the first differential
invariant d s ∗ 2 , namely the matrix of the metric G = diag [ r 2 cos 2 v, r 2 ], we are led to the first
representation d s 2 of the first differential invariant, which is equivariant or form-invariant: d s 2 =
r 2 cos 2 v d u 2 + r 2 d v 2 = r 2 cos 2 v ∗ d u ∗ 2 + r 2 d v ∗ 2 =d s ∗ 2 . Indeed, we have shown that under the action
of the three-dimensional rotation group, namely R( α,β,γ )=R 1 ( α )R 2 ( β )R 3 ( γ ) , d s 2 =d s ∗ 2 .
Sixth, we accordingly identify the three Killing vectors

{ d u, d v }→{

{ e 1 , e 2 , e 3 } or [1,0,0], [0 , 1 , 0], and [0 , 0 , 1],

respectively—the symmetry of the sphere S

r .

Box 3.2 (Sphere. Killing vectors of symmetry, equivariance of the arc length under the action
of the special orthogonal group SO(3)).
Sphere parameterized in an equatorial frame of reference:

x ( u,v )= e 1 cos v cos u + e 2 cos v sin u + e 3 sin v. (3.29)

Sphere parameterized in an oblique frame of reference:

x ( u ∗ ,v ∗ )= e 1 ∗ cos v ∗ cos u ∗ + e 2 ∗ cos v ∗ sin u ∗ + e 3 ∗ sin v ∗ .

(3.30)

Action of the special orthogonal group SO(3):

R ∗ R=

R( α,β,γ ) ∈ SO(3) := { R ∈ SO(3) | | 3 , | R | =1 } ,

⎡ ⎤ ⎡ ⎤

e 1 ∗
e 2 ∗
e 3 ∗

e 1
e 2
e 3

⎣ ⎦ =R 1 ( α )R 2 ( β )R 3 ( γ ) ⎣ ⎦

⇔

⎡ ⎤ ⎡ ⎤

e 1
e 2
e 3

e 1 ∗
e 2 ∗
e 3 ∗

⎣ ⎦ =R 3 ( γ )R 2 ( β )R 1 ( α ) ⎣ ⎦ ,

(3.31)

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

+ e 3 ∗ (sin γ sin α +cos γ sin β cos α ) ,

−