“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 182

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 tan-
gential 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

A 1 ,A 2 , or a reference figure of a celestial body like the triaxial

ellipsoid E

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 .
A first set of maps is illustrated by the magic triangle that is depicted in Fig. 5.2 .Fromthe
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 dif-
ferential 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) Lam-
bert ' smap . 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?