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 )= 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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