13 “Sphere to cylinder”: pseudo-cylindrical projections
Mapping the sphere to a cylinder: pseudo-cylindrical projections. Sinusoidal pseudo-cylindrical mapping,
elliptic pseudo-cylindrical mapping, parabolic pseudo-cylindrical mapping, rectilinear pseudo-cylindrical
mapping. Jacobi matrix, Cauchy–Green matrix, principal stretches.
Pseudo-cylindrical projections have, in the normal aspect, straight parallel lines for parallels. The
meridians are most often equally spaced along parallels, as they are on a cylindrical projection, but
on which the meridians are curved. Meridians may be mapped as straight lines or general curves.
13-1 General mapping equations
General mapping equations and distortion measures for pseudo-cylindrical mappings of the sphere. Jacobi
matrix, Cauchy–Green matrix, principal stretches.
The mapping equations are of the general form (13.1). The left Jacobi matrix is provided by (13.2)
and the left Cauchy–Green matrix (G r = I 2 ) by (13.3).
x = x(Λ, Φ) = RΛ cos Φg(Φ) ,
y = y(Φ) = Rf (Φ) ,
(13.1)
J l :=
D Λ x D Φ x
D Λ y D Φ y
=
= R
g(Φ) cos Φ −Λ[g(Φ) sin Φ − g
(Φ) cos Φ]
0
f
(Φ)
,
(13.2)
C l = J
∗
l G r J l =
= R
2
⎡
⎣
g
2 cos
2 Φ
−Λg
2 sin Φ cos Φ + Λg
g cos
2 Φ
−Λg
2 sin Φ cos Φ + Λg
g cos
2 Φ Λ
2 g
2 sin
2 Φ + f
2 + Λ
2 g
2 cos
2 Φ − 2Λ
2 gg
sin Φ cos Φ
⎤
⎦ .
(13.3)
The left principal stretches are determined from the characteristic equation det [C l − Λ
2
S G l ] = 0 and
G l = diag [R
2 cos
2 Φ, R
2 ], which leads to the biquadratic equation (13.4), the solution of which is
provided by (13.5).
Λ
4
S − Λ
2
S (Λ
2 g
2 sin
2 Φ + f
2 + g
2 + Λ
2 g
2 cos
2 Φ − 2Λ
2 gg
sin Φ cos Φ) + g
2 f
2 = 0 , (13.4)
Λ
2
S =
1
2 (Λ
2 g
2 sin
2 Φ + f
2 + g
2 + Λ
2 g
2 cos
2 Φ − 2Λ
2 gg
sin Φ cos Φ)±
±
1
4 (Λ 2 g 2 sin
2 Φ + f 2 + g 2 + Λ 2 g 2 cos 2 Φ − 2Λ 2 gg sin Φ cos Φ) − g 2 f 2 =:
=: a ± b .
(13.5)