“Sphere to Cone”: Polar Aspect - Map Projections: Cartographic Information Systems - page 424

17
“Sphere to Cone”: Polar Aspect

Mapping the sphere to a cone: polar aspect. Equidistant, conformal, and equal area mappings.
Ptolemy, de L'Isle, Lambert, and Albers projections. Point-like North Pole. Tangent cones,
secant cones, and circles-of-contact.

For mapping regional areas of medium latitude, conic mappings are particularly adequate (com-
pare with Fig. 17.1 ). The characteristic feature of conic mappings is that in the polar aspect
meridians are represented by straight lines which intersect in one point, the apex . Parallels are
mapped onto arcs of equicentric circles with the apex as the central point. As with cylindrical
mappings, there exist two cases: first, the cone touches the sphere along a parallel circle (com-
pare with Fig. 17.2 , top) and, second, it intersects the sphere along two parallels (compare with
Fig. 17.2 , bottom). Both cases are driven by the opening angle Θ ∈ (0 ,π/ 2), which is the vertex
angle made by a cross section through the apex and center of the base (compare with Fig. 17.3 ).

17-1 General Mapping Equations

Setting up general equations of the mapping “sphere to cone”: projections in the polar aspect.
Jacobi matrix, Cauchy-Green matrix, principal stretches.

The axis of the cone coincides with the polar axis of the Earth, i.e. the straight line passing
through the North Pole N and the center of the sphere. The main construction principals

are that first two points of equal spherical latitude Φ have the same distance r 0 from the map
center, which is the image of the apex. Second, the cone is sliced along the image of that meridian
which is diametrically opposed to the image of the central meridian (compare with Fig. 17.4 ).
Third, the cone can be developed into the plane. The circle-of-contact is that parallel circle
Φ = Φ 0 where the cone touches the sphere. If necessary, the cone is shifted along the polar
axis until the touching position is reached. The radius R 0 of the circle-of-contact is given by
R 0 = R cos Φ 0 . The slant height r 0 , which is the radius of the map image of the circle-of-contact,
is r 0 = R 0 / sin Φ 0 = R cot Φ 0 .

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