Geography Reference
In-Depth Information
Question 1: “Let a transformation group act on the coordi-
nate transformation of a surface-of-revolution. Indeed, we
make a coordinate transformation. What are the trans-
formation groups (the coordinate transformations) which
leave the
first differential invariant
d
s
2
of a surface-of-
revolution
equivariant
or
form-invariant
?” Answer 1: “The
transformation group, which leaves the first differential
invariant d
s
2
(also called “arc length”) equivariant is the
one-dimensional rotation group R
3
(
Ω
), a rotation around
the 3 axis of the ambient space
3
,δ
ij
{
R
}
.The3axisestab-
lishes the Killing vector of symmetry.”.
A proof of our answer is outlined in Box
3.1
. First, we present a parameter representation of a
surface-of-revolution, defined by
u
∗
,v
∗
}
in a rotated equatorial frame of reference. Second, we follow the action of the rotation group
R
3
(
Ω
)
{
u,v
}
in an equatorial frame of reference and defined by
{
∈
SO(2). Third, we generate the forward and backward transformations
{
e
1
, e
2
, e
3
|O} →
{
of orthonormal base vectors, which span the
three-dimensional Euclidean ambient space. Fourth, we then fill in the backward transformation
of bases into the first parameter representation of the surface-of-revolution and compare with the
second one. In this way, we find the “Kartenwechsel” (“cha-cha-cha”)
{u
∗
=
u−Ω, v
∗
=
v}
. Fifth,
we compute the first differential invariant d
s
2
of the surface-of-revolution, namely the matrix of
the metric G = diag[
f
2
,f
2
+
g
2
]. Cha-cha-cha leads us via the Jacobi map J to the second
representation d
s
∗
2
of the first differential invariant, which turns out to be equivariant or form-
invariant. Indeed, we have shown that under the action of the rotation group: d
s
2
=d
s
∗
2
. Sixth,
we identify
e
3
or [0
,
0
,
1] as the Killing vector of the symmetry of a surface-of-revolution.
e
1
∗
, e
2
∗
, e
3
∗
|O}
and
{
e
1
∗
, e
2
∗
, e
3
∗
|O} → {
e
1
, e
2
, e
3
|O}
Box 3.1 (Surface-of-revolution. Killing vector of symmetry, equivariance of the arc length
under the action of the special orthogonal group SO(2)).
Surface-of-revolution parameterized in an equatorial frame of reference:
x
(
u,v
)=
e
1
f
(
v
)cos
u
+
e
2
f
(
v
)sin
u
+
e
3
g
(
v
)
.
(3.21)
Surface-of-revolution parameterized in a rotated equatorial frame of reference:
x
(
v
∗
,v
∗
)=
e
1
∗
f
(
v
∗
)cos
u
∗
+
e
2
∗
f
(
v
∗
)sin
u
∗
+
e
3
∗
g
(
v
∗
)
.
(3.22)
Action of the special orthogonal group SO(2):
3
×
3
|
R
3
R
3
=
|
3
, |
R
3
|
=1
},
R
3
(
Ω
)
∈
SO(2) :=
{
R
3
∈
R
⎡
⎤
⎡
⎤
⎡
⎤
⎡
⎤
e
1
∗
e
2
∗
e
3
∗
e
1
e
2
e
3
cos
Ω
sin
Ω
0
−
e
1
e
2
e
3
⎣
⎦
=R
3
(
Ω
)
⎣
⎦
=
⎣
⎦
⎣
⎦
,
sin
Ω
cos
Ω
0
0
0 1
(3.23)
⎡
⎤
⎡
⎤
⎡
⎤
⎡
⎤
e
1
e
2
e
3
e
1
∗
e
2
∗
e
3
∗
cos
Ω
sin
Ω
0
sin
Ω
cos
Ω
0
0
−
e
1
∗
e
2
∗
e
3
∗
⎣
⎦
=R
3
(
Ω
)
⎣
⎦
=
⎣
⎦
⎣
⎦
,
0
1
e
1
=
e
1
∗
cos
Ω
−
e
2
∗
sin
Ω,
e
2
=
e
1
∗
sin
Ω
+
e
2
∗
cos
Ω,
e
3
=
e
3
∗
.
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