Geography Reference
In-Depth Information
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 map-
ping, 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
)
,
(13.1)
y
=
y
(
Φ
)=
Rf
(
Φ
)
,
J
l
:=
D
Λ
xD
Φ
x
=
(13.2)
D
Λ
yD
Φ
y
=
R
g
(
Φ
)cos
Φ
,
g
(
Φ
)cos
Φ
]
−
Λ
[
g
(
Φ
)sin
Φ
−
f
(
Φ
)
0
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