Geography Reference
In-Depth Information
H
Generalized Hammer Projection
Generalized Hammer projection of the ellipsoid-of-revolution: azimuthal, transverse, rescaled
equiareal. Mapping equations. Univariate series inversion.
The
classical Hammer projection
of the
sphere
, which is azimuthal, transverse rescaled equiareal, is
generalized
to the
ellipsoid-of-revolution
. Its first constituent, the azimuthal transverse equiareal
projection of the biaxial ellipsoid, is derived giving the equations for an equiareal transverse
azimuthal projection. The second constituent, the equiareal mapping of the biaxial ellipsoid with
respect to a transverse frame of reference and a change of scale, is reviewed. Then considered
results give collections of the general mapping equations generating the ellipsoidal Hammer pro-
jection, which finally lead to a world map.
One of the most widely used equiareal map projection is the
Hammer projection
of the
sphere
(
Hammer 1892
). It maps
parallel circles and meridians of the sphere onto algebraic
curves of fourth order; its limit line (
Λ
=
π
) is an ellipse
with respect to the gauge
c
1
=2
,c
2
=1
,c
3
=1
/
2, and
c
4
=
1 as illustrated by Fig.
H.1
with respect to the mapping
equations (
H.87
), (
H.88
), and (
H.96
), respectively. Many
celestial bodies like the Earth are pronounced
ellipsoidal
.
It is therefore our target to generalize the spherical Hammer
projection to an ellipsoid-of-revolution to which we refer as
a biaxial ellipsoid.
±
Section H-1, Section H-3, Section H-4.
The first constituent of the Hammer projection is the transverse equiareal projection onto a
tangent plane, namely of azimuthal type. Section
H-1
outlines accordingly the introduction
of a transverse reference frame for a biaxial ellipsoid. In particular, formulae in (
H.12
)are
derived which constitute the transformation from surface normal ellipsoidal longitude/latitude
{
Λ
∗
,Φ
∗
}
A
∗
,B
∗
}
defined in the ellip-
soidal transverse frame of reference. In consequence, the transverse equiareal mapping of a
biaxial ellipsoid onto a transverse tangent plane is given by corollaries in terms of ellipsoidal
to surface normal ellipsoidal meta-longitude/meta-latitude
{
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