Geography Reference
In-Depth Information
E-34 A Geodesic as a Submanifold of the Ellipsoid-of-Revolution
(Conformal Coordinates)
Let us represent the differential equations of a geodesic as a submanifold of the ellipsoid-of-
revolution
A,B
as functions
q
μ
(
t
) as well as the Lagrangean
L
2
and the Hamiltonian
H
2
with respect to conformal coordinates (isothermal, isometric) of the most important appli-
cable map projections listed below. Compare with Fig.
E.5
, which illustrates the Maupertuis
gauged geodesic
2
:=
2
M
E
2
A,B
generated by the Universal Transverse Mercator projection (UTM) with
the dilatation factor
ρ
=0
.
999578 (E. Grafarend 1995). Compare with Boxes
E.4
-
E.6
,which
collect the central relations.
E
(i) The Universal Polar Stereographic Projection (UPS)
(central perspective projection from the South Pole to a
tangent plane
T
np
2
at the North Pole), (ii) the Uni-
versal Mercator Projection (UM) (conformal diffeomor-
phism of the ellipsoid-of-revolution onto a circular cylin-
der
S
M
A
is chosen as the equator of
the ellipsoid-of-revolution), (iii) the Universal Transverse
Mercator Projection (UTM) (conformal diffeomorphism of
the ellipsoid-of-revolution onto a elliptic cylinder (
Λ
0
meta-
equator), where the
Λ
0
ellipse is chosen as the reference
meridian of the ellipsoid-of-revolution), and (iv) the Uni-
versal Conic Lambert Projection (UC) (conformal diffeo-
morphism of the ellipsoid-of-revolution onto a circular cone
{
A
×
R
where the circle
S
3
(
x
2
+
y
2
)
/a
2
z
2
/b
2
=0
,a
2
+
b
2
=const
.
x
∈
R
|
−
}
,where
e
2
sin
2
Φ
)
−
1
/
2
is chosen
as a definite parallel circle (line-of-contact) and
b
as the lat-
itude dependent definite distance of the circle
1
the circle
S
a
with
a
=
A
cos
Φ/
(1
−
1
S
a
from the
equatorial plane, and
e
2
=(
A
2
B
2
)
/A
2
=1
(
B
2
/A
2
)).
−
−
Box E.4 (The conformal coordinates as functions of ellipsoidal longitude and ellipsoidal
latitude and the factor of conformality).
(i) Universal Polar Stereographic Projection (UPS) :
q
1
=
f
(
Φ
)cos
Λ, q
2
=
f
(
Φ
)sin
Λ,
(E.68)
1
2
1+
e
sin
Φ
1
2
tan
π
,
2
A
√
1
e
1+
e
−
Φ
2
f
(
Φ
):=
4
−
(E.69)
−
e
sin
Φ
−
e
2
A
cos
f
−
1
(
q
1
)
2
+(
q
2
)
2
)
1
A
cos
Φ
1
fΦ
=
1
λ
=
e
2
sin
2
f
−
1
(
q
1
)
2
+(
q
2
)
2
e
2
sin
2
Φ
−
−
Search WWH ::
Custom Search