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 projection, which finally lead to
a world map.
Aside.
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
∗
} to surface normal
ellipsoidal meta-longitude/meta-latitude {A
∗ , B
∗
} defined in the ellipsoidal 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 meta-longitude/ meta-latitude. Finally, the elaborate
transformation {Λ
∗ , Φ
∗
} → {A
∗ , B
∗
} in these transverse equiareal mapping equations is performed,
leading to the final result of Lemma H.3 and Corollary H.4. Mathematical details are collected in
Sections H-3 and H-4
Section H-2, Section H-5.
The second constituent of the Hammer projection is an alternative equiareal mapping of the biaxial ellipsoid with respect to a transverse frame of reference and a change of scale outlined in Section H-2. As a
reference, we firstly present an equiareal mapping from a left biaxial ellipsoid to a right biaxial ellipsoid.
We secondly give the explicit form of the mapping equations generating an equiareal map from a left biaxial ellipsoid to a right biaxial ellipsoid with respect to a transverse frame of reference and a change of
scale. The general mapping equations x = c 1 x
∗ (Λ
∗ , Φ
∗ ) and y = c 2 y
∗ (Λ
∗ , Φ
∗ ) are specified by means of
the differential equations dΛ
∗ /dΛ = c 3 and dΦ
∗ /dΦ = c 4 (1−E
2 sin
2 Φ
∗ )
2 cos Φ/[(1−E
2 sin
2 Φ)
2 cos Φ
∗ ]
subject to the initial data {Λ
∗ (Λ = 0) = 0, Φ
∗ (Φ = 0) = 0} of left and right longitude/latitude. In
order to guarantee an equiareal mapping, the constants of gauge have to fulfill c 1 c 2 c 3 c 4 = 1. Termwise
integration of the differential equation dΦ
∗ /dΦ leads to the problem to determine sin Φ
∗ from a homogeneous polynomial equation which is solved by univariate series inversion in Section H-5. Third, the
general equations of the ellipsoidal Hammer projection are presented in Lemma H.5 and made more
specififc by Corollary H.6, namely with respect to the Hammer gauge {c 1 = 2, c 2 = 1, c 3 = 1/2, c 4 = 1}
which guarantees a map of the total left biaxial ellipsoid onto one chart. Finally, Fig. H.3 presents a
world map of the ellipsoidal Hammer projection for a relative eccentricity E = 0.
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 projection, which finally lead to
a world map.
Aside.
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
∗
} to surface normal
ellipsoidal meta-longitude/meta-latitude {A
∗ , B
∗
} defined in the ellipsoidal 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 meta-longitude/ meta-latitude. Finally, the elaborate
transformation {Λ
∗ , Φ
∗
} → {A
∗ , B
∗
} in these transverse equiareal mapping equations is performed,
leading to the final result of Lemma H.3 and Corollary H.4. Mathematical details are collected in
Sections H-3 and H-4
Section H-2, Section H-5.
The second constituent of the Hammer projection is an alternative equiareal mapping of the biaxial ellipsoid with respect to a transverse frame of reference and a change of scale outlined in Section H-2. As a
reference, we firstly present an equiareal mapping from a left biaxial ellipsoid to a right biaxial ellipsoid.
We secondly give the explicit form of the mapping equations generating an equiareal map from a left biaxial ellipsoid to a right biaxial ellipsoid with respect to a transverse frame of reference and a change of
scale. The general mapping equations x = c 1 x
∗ (Λ
∗ , Φ
∗ ) and y = c 2 y
∗ (Λ
∗ , Φ
∗ ) are specified by means of
the differential equations dΛ
∗ /dΛ = c 3 and dΦ
∗ /dΦ = c 4 (1−E
2 sin
2 Φ
∗ )
2 cos Φ/[(1−E
2 sin
2 Φ)
2 cos Φ
∗ ]
subject to the initial data {Λ
∗ (Λ = 0) = 0, Φ
∗ (Φ = 0) = 0} of left and right longitude/latitude. In
order to guarantee an equiareal mapping, the constants of gauge have to fulfill c 1 c 2 c 3 c 4 = 1. Termwise
integration of the differential equation dΦ
∗ /dΦ leads to the problem to determine sin Φ
∗ from a homogeneous polynomial equation which is solved by univariate series inversion in Section H-5. Third, the
general equations of the ellipsoidal Hammer projection are presented in Lemma H.5 and made more
specififc by Corollary H.6, namely with respect to the Hammer gauge {c 1 = 2, c 2 = 1, c 3 = 1/2, c 4 = 1}
which guarantees a map of the total left biaxial ellipsoid onto one chart. Finally, Fig. H.3 presents a
world map of the ellipsoidal Hammer projection for a relative eccentricity E = 0.