5 “Sphere to tangential plane”: polar (normal) aspect
Mapping the sphere to a tangential plane: polar (normal) aspect. Equidistant, conformal, and equal area
mappings. Normal perspective mappings. Pseudo-azimuthal mapping. Wiechel polar pseudo-azimuthal
mapping. Northern tangential plane, equatorial plane, southern tangential plane. Gnomonic and orthographic projections. Lagrange projection.
For mapping local and regional areas, maps of a surface (for instance, a topographic surface TOP,
a reference figure of a celestial body like the sphere S
2
R , a reference figure of a celestial body like
the ellipsoid-of-revolution E
2
A 1 ,A 2
, or a reference figure of a celestial body like the triaxial ellipsoid
E
2
A 1 ,A 2 ,A 3
) onto a tangential plane are without competition. In this introductory chapter, we focus
on mapping the sphere to a tangential plane, which is located either at the North Pole or at the
South Pole. Such a placement of the plane “we map onto” is conventionally called polar aspect. Since
the spherical coordinate Λ coincides with the polar coordinate α of a point in the tangent plane, the
mapping is called azimuthal mapping: α = Λ. Later on, we generalize from the polar aspect to the
transverse aspect, finally to the oblique aspect. For a first impression, consult Fig. 5.1.
Fig. 5.1. Mapping the sphere to a tangential plane: polar aspect. Point-of-contact: North Pole. Parameters:
Λ 0 ∈ [0
◦ , 360
◦ ], Φ 0 = 90
◦ .
A first set of maps is illustrated by the magic triangle that is depicted in Fig. 5.2. From the
canonical postulates of principal stretches (i) Λ 2 = 1, (ii) Λ 1 = Λ 2 , and (iii) Λ 1 Λ 2 = 1, we generate
the differential equations which characterize (i) an equidistant mapping, (ii) a conformal mapping
(conformeomorphism), and (iii) an equiareal mapping (areomorphism). These characteristic differential
equations are uniquely solved with respect to a properly chosen initial value. The related maps are
called (i) Postel’s map, (ii) Universal Polar Stereographic (UPS) map, and (iii) Lambert’s map. In
addition, we produce a second set of maps called normal perspective. We identify the perspective
center, the line-of-sight, and the line-of-contact, and we discuss the minimal and complete atlas. The
guided tour through the world of azimuthal projective maps brings us to special maps, which are
called (i) the gnomonic projection, (ii) the orthographic projection, and (iii) the Lagrange projection,
and which are pointed out by Fig. 5.3. Finally, we answer the key question: What are the best polar
azimuthal mappings of the sphere to the plane?
Historical
aside.
Note that the gnomonic projection is believed to has been invented by Thales of Milet
(1st half of 6th century B.C.), the equiareal azimuthal projection has been invented by
J. H. Lambert (∗ 26 August 1728, Muelhausen, Elsass; † 25 September 1777, Berlin), and
G. Postel used the equidistant azimuthal projection for a first map of France (1568, 1570).
Mapping the sphere to a tangential plane: polar (normal) aspect. Equidistant, conformal, and equal area
mappings. Normal perspective mappings. Pseudo-azimuthal mapping. Wiechel polar pseudo-azimuthal
mapping. Northern tangential plane, equatorial plane, southern tangential plane. Gnomonic and orthographic projections. Lagrange projection.
For mapping local and regional areas, maps of a surface (for instance, a topographic surface TOP,
a reference figure of a celestial body like the sphere S
2
R , a reference figure of a celestial body like
the ellipsoid-of-revolution E
2
A 1 ,A 2
, or a reference figure of a celestial body like the triaxial ellipsoid
E
2
A 1 ,A 2 ,A 3
) onto a tangential plane are without competition. In this introductory chapter, we focus
on mapping the sphere to a tangential plane, which is located either at the North Pole or at the
South Pole. Such a placement of the plane “we map onto” is conventionally called polar aspect. Since
the spherical coordinate Λ coincides with the polar coordinate α of a point in the tangent plane, the
mapping is called azimuthal mapping: α = Λ. Later on, we generalize from the polar aspect to the
transverse aspect, finally to the oblique aspect. For a first impression, consult Fig. 5.1.
Fig. 5.1. Mapping the sphere to a tangential plane: polar aspect. Point-of-contact: North Pole. Parameters:
Λ 0 ∈ [0
◦ , 360
◦ ], Φ 0 = 90
◦ .
A first set of maps is illustrated by the magic triangle that is depicted in Fig. 5.2. From the
canonical postulates of principal stretches (i) Λ 2 = 1, (ii) Λ 1 = Λ 2 , and (iii) Λ 1 Λ 2 = 1, we generate
the differential equations which characterize (i) an equidistant mapping, (ii) a conformal mapping
(conformeomorphism), and (iii) an equiareal mapping (areomorphism). These characteristic differential
equations are uniquely solved with respect to a properly chosen initial value. The related maps are
called (i) Postel’s map, (ii) Universal Polar Stereographic (UPS) map, and (iii) Lambert’s map. In
addition, we produce a second set of maps called normal perspective. We identify the perspective
center, the line-of-sight, and the line-of-contact, and we discuss the minimal and complete atlas. The
guided tour through the world of azimuthal projective maps brings us to special maps, which are
called (i) the gnomonic projection, (ii) the orthographic projection, and (iii) the Lagrange projection,
and which are pointed out by Fig. 5.3. Finally, we answer the key question: What are the best polar
azimuthal mappings of the sphere to the plane?
Historical
aside.
Note that the gnomonic projection is believed to has been invented by Thales of Milet
(1st half of 6th century B.C.), the equiareal azimuthal projection has been invented by
J. H. Lambert (∗ 26 August 1728, Muelhausen, Elsass; † 25 September 1777, Berlin), and
G. Postel used the equidistant azimuthal projection for a first map of France (1568, 1570).