Preface VII
In contrast, Chapters 14–16 are a review in mapping an ellipsoid-of-revolution to a cylinder. We
start with the polar aspect of type {x = AΛ, y = f (Φ)}, specialize to normal equidistant, normal conformal, and normal equiareal, in general, to a rotationally symmetric figure (for example,
the torus). The transverse aspect is applied to the transverse Mercator projection and the special
Gauss–Krueger coordinates (UTM, GK) derived from the celebrated Korn–Lichtenstein equations subject to an integrability condition and an optimality condition for estimating the factor of conformality
(dilatation factor) in a given quantity range [−l E , +l E ] × [B S , B N ] = [−3.5
◦ , +3.5
◦ ] × [80
◦ S, 84
◦ N] or
[−l E , +l E ] × [B S , B N ] = [−2
◦ , +2
◦ ] × [80
◦ S, 80
◦ N], namely ρ = 0.999, 578 or ρ = 0.999, 864. Due to its
practical importance, we have added three examples for the transverse Mercator projection and for
the Gauss–Krueger coordinate system of type {Easting, Northing}, adding the meridian zone number.
Another special topic is the strip transformation from one meridian strip system to another one, both
for Gauss–Krueger coordinates and for UTM coordinates. We conclude with two detailed examples of
strip transformation (Bessel ellipsoid, World Geodetic System 84). At the end, we present to you the
oblique aspect of type Oblique Mercator Projection (UOM) of the ellipsoid-of-revolution, also called
rectified skew orthomorphic by M. Hotine. J. P. Snyder calls it “Hotine Oblique Mercator Projection
(HOM)”. Landsat-type data are a satellite example.
Only in the polar aspect, we present in Chapter 17 the maps of the sphere to the cone. We use Fig. 17.1
as an illustration and the setup {a = Λ sin Φ 0 , r = f (Φ)} in terms of polar coordinates. n := sin Φ 0
range from n = 0 for the cylinder to n = 1 for the azimuthal mapping. Thus, we are left with the
rule 0 < n < 1 for conic projections. The wide variety of conic projections were already known to
Ptolemy as the equidistant and conformal version on the circle-of-contact. If we want a point-like image
of the North Pole, the equidistant and conformal version on the circle-of-contact is our favorite.
Another equidistant and conformal version on two parallels is the de L’Isle mapping. Various versions
of conformal mapping range from the equidistant mappings on the circle-of-contact to the equidistant
mappings on two parallels (secant cone, J. H. Lambert). The equal area mappings range from the case
of an equidistant and conformal mapping on the circle-of-contact over the case of an equidistant and
conformal mapping on the circle-of-contact and a point-like image of the North Pole to the case of
equidistance and conformality on two parallels (secant cone, H. C. Albers).
Chapter 18 is an introduction into mapping the sphere to the cone, namely of type pseudo-conic. We
specialize on the Stab–Werner projection and on the Bonne projection. Both types have the shape of
the heart.
The polar aspect of mapping the ellipsoid-of-revolution to the cone is the key topic of Chapter 19. We
review the line-of-contact and the principal stretches before we enter into special cases, namely of type
equidistant mappings on the set of parallel circles of type conformal (variant equidistant on the circleof-reference, variant equidistant on two parallel circles, generalized Lambert conic projection) and type
equal area (variant equidistant and conformal on the reference circle, variant pointwise mapping of
the central point and equidistant and conformal on the parallel circle, variant of an equidistant and
conformal mapping on two parallel circles, generalized Albers conic projection).
In contrast, Chapters 14–16 are a review in mapping an ellipsoid-of-revolution to a cylinder. We
start with the polar aspect of type {x = AΛ, y = f (Φ)}, specialize to normal equidistant, normal conformal, and normal equiareal, in general, to a rotationally symmetric figure (for example,
the torus). The transverse aspect is applied to the transverse Mercator projection and the special
Gauss–Krueger coordinates (UTM, GK) derived from the celebrated Korn–Lichtenstein equations subject to an integrability condition and an optimality condition for estimating the factor of conformality
(dilatation factor) in a given quantity range [−l E , +l E ] × [B S , B N ] = [−3.5
◦ , +3.5
◦ ] × [80
◦ S, 84
◦ N] or
[−l E , +l E ] × [B S , B N ] = [−2
◦ , +2
◦ ] × [80
◦ S, 80
◦ N], namely ρ = 0.999, 578 or ρ = 0.999, 864. Due to its
practical importance, we have added three examples for the transverse Mercator projection and for
the Gauss–Krueger coordinate system of type {Easting, Northing}, adding the meridian zone number.
Another special topic is the strip transformation from one meridian strip system to another one, both
for Gauss–Krueger coordinates and for UTM coordinates. We conclude with two detailed examples of
strip transformation (Bessel ellipsoid, World Geodetic System 84). At the end, we present to you the
oblique aspect of type Oblique Mercator Projection (UOM) of the ellipsoid-of-revolution, also called
rectified skew orthomorphic by M. Hotine. J. P. Snyder calls it “Hotine Oblique Mercator Projection
(HOM)”. Landsat-type data are a satellite example.
Only in the polar aspect, we present in Chapter 17 the maps of the sphere to the cone. We use Fig. 17.1
as an illustration and the setup {a = Λ sin Φ 0 , r = f (Φ)} in terms of polar coordinates. n := sin Φ 0
range from n = 0 for the cylinder to n = 1 for the azimuthal mapping. Thus, we are left with the
rule 0 < n < 1 for conic projections. The wide variety of conic projections were already known to
Ptolemy as the equidistant and conformal version on the circle-of-contact. If we want a point-like image
of the North Pole, the equidistant and conformal version on the circle-of-contact is our favorite.
Another equidistant and conformal version on two parallels is the de L’Isle mapping. Various versions
of conformal mapping range from the equidistant mappings on the circle-of-contact to the equidistant
mappings on two parallels (secant cone, J. H. Lambert). The equal area mappings range from the case
of an equidistant and conformal mapping on the circle-of-contact over the case of an equidistant and
conformal mapping on the circle-of-contact and a point-like image of the North Pole to the case of
equidistance and conformality on two parallels (secant cone, H. C. Albers).
Chapter 18 is an introduction into mapping the sphere to the cone, namely of type pseudo-conic. We
specialize on the Stab–Werner projection and on the Bonne projection. Both types have the shape of
the heart.
The polar aspect of mapping the ellipsoid-of-revolution to the cone is the key topic of Chapter 19. We
review the line-of-contact and the principal stretches before we enter into special cases, namely of type
equidistant mappings on the set of parallel circles of type conformal (variant equidistant on the circleof-reference, variant equidistant on two parallel circles, generalized Lambert conic projection) and type
equal area (variant equidistant and conformal on the reference circle, variant pointwise mapping of
the central point and equidistant and conformal on the parallel circle, variant of an equidistant and
conformal mapping on two parallel circles, generalized Albers conic projection).