“Ellipsoid-of-Revolution to Cylinder”: Oblique Aspect - Map Projections: Cartographic Information Systems - page 403

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“Ellipsoid-of-Revolution to Cylinder”: Oblique Aspect

Mapping the ellipsoid-of-revolution to a cylinder: oblique aspect. Oblique Mercator Projec-
tion (UOM), rectified skew orthomorphic projections. Korn-Lichtenstein equations, Laplace-
Beltrami equations.

In the world of conformal mappings of the Earth or other celestial bodies, the Mercator projection
plays a central role. The Mercator projection of the sphere

2

S

r or of the ellipsoid-of-revolution

2
A 1 ,A 2

E beside conformality is characterized by the equidistant mapping of the equator. In con-

trast, the transverse Mercator projection is conformal and maps the transverse meta-equator, the
meridian of reference, equidistantly. Accordingly, the Mercator projection is very well suited for
regions which extend East-West around the equator, while the transverse Mercator projection
fits well to those regions which have a South-North extension. Obviously, several geographical
regions are centered along lines which are neither equatorial, parallel circles, or meridians, but
may be taken as central intersection of a plane and the reference figure of the Earth or other
celestial bodies, the ellipsoid-of-revolution (spheroid). For geodetic applications, conformality is
desired in such cases, the Universal Oblique Mercator Projection (UOM) is the projection which
should be chosen. A study of the conformal projection of the ellipsoid- of-revolution by Hotine
( 1946 , 1947a , b , c , d ) is the basis of the ellipsoidal oblique Mercator projection, which M. Hotine
called the “rectified skew orthomorphic”, mainly applied in the United States (e.g. for Alaska), for
Malaysia, and for Borneo Hotine ( 1947a , b , c , d ), for the sphere by Laborde Chef d'escadron ( 1928 )
for Madagascar, by Rosenmund ( 1903 ) for Switzerland and by Cole ( 1943 ) for Italy, namely in
the context of the celebrated Gauss double projection (conformal mapping of the ellipsoid-of-
revolution to the sphere and of the sphere to the plane). According to Snyder ( 1982 ,p.76),the
Hotine Oblique mercator Projection (HOM) is the most suitable projection available for mapping
Landsat type data. HOM has also been used to cast the Heat Capacity Mapping Mission (HCMM)
imagery since 1978. Note that our interest in the Oblique Mercator was raised by the personally
obscure procedure to derive the mapping equations which should be based on similar concepts
known for Normal Mercator and Transverse Mercator. The mapping equations should guarantee
that the elliptic meta-equator should be mapped equidistantly. Accordingly, we derive here the
general mapping equations x ( L, B )and y ( L, B ) for conformal coordinates (isometric coordinates,
isothermal coordinates) as a function of ellipsoidal longitude L and ellipsoidal latitude B ,which
map the line-of-intersection (an ellipse) of an inclined central plane and the ellipsoid-of-revolution
equidistantly.