VI Preface
The standard map projections: tangential, cylindric, conic.
The Chapters 5–7 on mapping the sphere to the tangential plane, namely in the polar aspect (normal
aspect) – for instance, the Universal Polar Stereographic Projection (UPS) – and the meta-azimuthal
mapping in the transverse as well as the oblique aspect, follow. They range from equidistant mapping
via conformal mapping to equal area mapping, finally to normal perspective mappings. Special cases
are mappings of type “sphere to tangential plane” at maximal distance, at minimal distance, and at
the equatorial plane (three cases). We treat the line-of-sight, the line-of-contact, and minimal versus
complete atlas. The gnomonic projection, the orthographic projection, and the Lagrange projection
follow. Finally, we ask the question: “what is the best projection in the class of polar and azimuthal
projections of the sphere to the plane?” A special section on pseudoazimuthal mappings, namely the
Wiechel polar pseudoazimuthal mapping, and another special section on meta-azimuthal projections
(stereographic, transverse Lambert, oblique UPS and oblique Lambert) concludes the important chapter on various maps “sphere to plane”.
Chapter 8 is the first chapter on mapping the ellipsoid-of-revolution to the tangential plane. We treat
special mappings of type equidistant, conformal, and equal area, and of type perspective. Chapter 9
is the first chapter on double projections. First, we introduce the celebrated Gauss double projection.
Alternatively, we introduce the authalic equal area projection of the ellipsoid to the sphere and from
the sphere to the plane.
The four Chapters 10–13 are devoted to the mapping “sphere to cylinder”, namely to the polar aspect, to the meta-cylindric projections of type transverse and of type oblique, and finally to the
pseudo-cylindrical mode. Four examples, namely from mapping the sphere to a cylinder (polar aspect, transversal aspect, oblique aspect, pseudo-cylindrical equal area projections) in Chapters 10–13
document the power of these spherical projections. The resulting map projections are called (i)
Plate Carr´ ee (quadratische Plattkarte), (ii) Mercator projection (Gerardus Mercator 1512–1594), and
(iii) equal area Lambert projection. A special feature of the Mercator projection is its property “mapping loxodromes (rhumblines, lines of constant azimuths) to a straight line crossing all meridians with
a constant angle”. The most popular map projection is the Universal Transverse Mercator projection
(UTM) of the sphere to the cylinder, illustrated in Fig. 11.3. The pseudo-cylindrical equal area projections – they only exist – are widely used in the sinusoidal version (Cossin, Sanson–Flamsteed), in
the elliptic version (Mollweide, very popular), in the parabolic version (Craster), and in the rectilinear
version (Eckert II).
In Chapter 10, a special section is devoted to the question “what is the best cylindric projection when
best is measured by the Airy optimal criterion or by the Airy–Kavrajski optimal criterion?” We have
compared three mappings: (i) conformal, (ii) equal area, and (iii) distance preserving in the class of
“equidistance on two parallel circles”. We prove that the distance preserving maps are optimal and
the equal area maps are better than the conformal maps, at least until a latitude of Φ = 56
◦ , when we
apply the Airy optimal criterion. Alternatively, when we measure optimality by the Airy–Kavrajski
optimal criterion, we find again that the optimum is with the distance preserving maps, but conformal
maps produce exactly the same equal area maps, less optimal compared to distance preserving maps.
The standard map projections: tangential, cylindric, conic.
The Chapters 5–7 on mapping the sphere to the tangential plane, namely in the polar aspect (normal
aspect) – for instance, the Universal Polar Stereographic Projection (UPS) – and the meta-azimuthal
mapping in the transverse as well as the oblique aspect, follow. They range from equidistant mapping
via conformal mapping to equal area mapping, finally to normal perspective mappings. Special cases
are mappings of type “sphere to tangential plane” at maximal distance, at minimal distance, and at
the equatorial plane (three cases). We treat the line-of-sight, the line-of-contact, and minimal versus
complete atlas. The gnomonic projection, the orthographic projection, and the Lagrange projection
follow. Finally, we ask the question: “what is the best projection in the class of polar and azimuthal
projections of the sphere to the plane?” A special section on pseudoazimuthal mappings, namely the
Wiechel polar pseudoazimuthal mapping, and another special section on meta-azimuthal projections
(stereographic, transverse Lambert, oblique UPS and oblique Lambert) concludes the important chapter on various maps “sphere to plane”.
Chapter 8 is the first chapter on mapping the ellipsoid-of-revolution to the tangential plane. We treat
special mappings of type equidistant, conformal, and equal area, and of type perspective. Chapter 9
is the first chapter on double projections. First, we introduce the celebrated Gauss double projection.
Alternatively, we introduce the authalic equal area projection of the ellipsoid to the sphere and from
the sphere to the plane.
The four Chapters 10–13 are devoted to the mapping “sphere to cylinder”, namely to the polar aspect, to the meta-cylindric projections of type transverse and of type oblique, and finally to the
pseudo-cylindrical mode. Four examples, namely from mapping the sphere to a cylinder (polar aspect, transversal aspect, oblique aspect, pseudo-cylindrical equal area projections) in Chapters 10–13
document the power of these spherical projections. The resulting map projections are called (i)
Plate Carr´ ee (quadratische Plattkarte), (ii) Mercator projection (Gerardus Mercator 1512–1594), and
(iii) equal area Lambert projection. A special feature of the Mercator projection is its property “mapping loxodromes (rhumblines, lines of constant azimuths) to a straight line crossing all meridians with
a constant angle”. The most popular map projection is the Universal Transverse Mercator projection
(UTM) of the sphere to the cylinder, illustrated in Fig. 11.3. The pseudo-cylindrical equal area projections – they only exist – are widely used in the sinusoidal version (Cossin, Sanson–Flamsteed), in
the elliptic version (Mollweide, very popular), in the parabolic version (Craster), and in the rectilinear
version (Eckert II).
In Chapter 10, a special section is devoted to the question “what is the best cylindric projection when
best is measured by the Airy optimal criterion or by the Airy–Kavrajski optimal criterion?” We have
compared three mappings: (i) conformal, (ii) equal area, and (iii) distance preserving in the class of
“equidistance on two parallel circles”. We prove that the distance preserving maps are optimal and
the equal area maps are better than the conformal maps, at least until a latitude of Φ = 56
◦ , when we
apply the Airy optimal criterion. Alternatively, when we measure optimality by the Airy–Kavrajski
optimal criterion, we find again that the optimum is with the distance preserving maps, but conformal
maps produce exactly the same equal area maps, less optimal compared to distance preserving maps.