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
=
1
λ =
e 2 sin 2 f 1 ( q 1 ) 2 +( q 2 ) 2
e 2 sin 2 Φ
 
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