“Sphere to Cylinder”: Polar Aspect - Map Projections: Cartographic Information Systems - page 304

10
“Sphere to Cylinder”: Polar Aspect
Mapping the sphere to a cylinder: polar aspect. Equidistant, conformal, and equal area
mappings. Principle for constructing a cylindrical map projection. Optimal cylinder projec-
tions of the sphere, equidistant on two parallels.
In this chapter, we present a collection of most widely used map projections in the polar aspect
in which meridians are shown as a set of equidistant parallel straight lines and parallel circles
(parallels) by a system of parallel straight lines orthogonally crossing the images of the meridians.
As a specialty, the poles are not displayed as points but straight lines as long as the equator. First,
we derive the general mapping equations for both cases of (i) a tangent cylinder and (ii) a secant
cylinder and describe the construction principle. The mapping equations and the equations for
the left principal stretches involve a general latitude dependent function f , which is determined
in a following section through the postulate of (i) an equidistant, (ii) a conformal, or (iii) an
equal area mapping. The resulting map projection are the most simple Plate Carree projection
(“quadratische Plattkarte”), the famous conformal Mercator projection (presented by Gerardus
Mercator (Latinized name of Gerhard Kremer, 1512-1594) of Flanders in 1569) and the equal
area Lambert projection (presented by Johann Heinrich Lambert (1728-1777) of Alsace in 1772).
While the Plate Carree projection was mainly used for the representation of equatorial regions,
the Mercator projection has found widespread use in (aero-)nautics and maps for displaying air
and ocean currents. A special feature of this projection is that the loxodrome (rhumb line, line
of constant azimuth) is displayed as a straight line crossing all meridians with a constant angle.
The cylindrical Lambert projection, in contrast, has found only minimal usage, which is mainly
due to the fact that the images of parallels lie very dense in medium and high latitudes. For a
first impression, have a look at Fig. 10.1 .
10-1 General Mapping Equations
Setting up general equations of the mapping “sphere to cylinder”: projections in the polar
aspect. Principle for constructing a cylindrical map projection.