“Sphere to Cylinder”: Polar Aspect - Map Projections: Cartographic Information Systems - page 305

Fig. 10.1. Mapping the sphere to a (tangent) cylinder. Polar aspect. Line-of-contact: equator

There are two basic postulates which govern the setup of general equations of mapping the sphere
S

2 R . First, the coordinate x depends only on the
longitude Λ and the parallel circles Φ = ±Φ 0 have to be mapped equidistantly, i.e. x = RΛ cos Φ 0 .
Second, the coordinate y is only a function of latitude Φ , i.e. y = f ( Φ ), compare with Fig. 10.2 for
the case of a tangent cylinder. In case of the tangent variant, the cylinder is wrapping the sphere
with the equator being the line-of-contact. In the second case of a secant cylinder, two parallel
circles Φ = ±Φ 0 are the lines-of-contact, compare with Fig. 10.3 .

2 R of radius R to a tangent or secant cylinder

C

Box 10.1 (“Sphere to cylinder”: distortion analysis, polar aspect, left principal stretches).
Parameterized mapping:

x = RΛ cos Φ 0 , y = f ( Φ ) .

(10.1)

Left Jacobi matrix:
J l := D Λx D Φx

= R cos Φ 0 0 .

(10.2)

D Λ yD Φy 0 f ( Φ )