290 12 “Sphere to cylinder”: oblique aspect
12-1 General mapping equations
Setting up general equations of the mapping “sphere to cylinder”: projections in the oblique aspect.
Meta-longitude, meta-latitude.
The general equations for mapping the sphere to a cylinder in the transverse aspect are based on the
general equation (10.1) of Chapter 10, but spherical longitude Λ and spherical latitude Φ being replaced
by their counterparts meta-longitude and meta-latitude, which are indicated here by capital letters
A and B. In order to treat simultaneously both the transverse tangent cylinder and the transverse
secant cylinder, we introduce B 0 as the meta-latitude of the meta-parallel circles B = ±B 0 which shall
be mapped equidistantly. In consequence, the general equations for this case are given by the very
general vector relation (12.1), taking into account the constraints (3.51) and (3.53), namely (12.2).
For the distortion analysis, the left principal stretches result to (12.3).
x
y
=
RA cos B 0
f (B)
,
(12.1)
tan A =
cos Φ sin(Λ − Λ 0 )
cos Φ sin Φ 0 cos(Λ − Λ 0 ) − sin Φ cos Φ 0
,
sin B = cos Φ cos Φ 0 cos(Λ − Λ 0 ) + sin Φ sin Φ 0 ,
(12.2)
Λ 1 =
cos B 0
cos B
, Λ 2 =
f
(B)
R
.
(12.3)
The procedure of how to set up special equations of the mapping sphere to cylinder in the oblique
aspect (oblique equidistant mapping, oblique conformal mapping, oblique equal area mapping)
can be easily deduced from the preceding chapters. Far easier, in the mapping equations as well
in the equations for the left principal stretches defined in Chapter 10, conventional coordinates
spherical longitude Λ and spherical latitudes Φ and Φ 0 are simply replaced by their corresponding
items meta-spherical longitude A and meta-spherical latitudes B and B 0 . Transformations of conventional spherical coordinates to meta-spherical coordinates is performed using (12.2).
12-2 Special mapping equations
Setting up special equations of the mapping “sphere to cylinder”: meta-cylindrical projections in the
oblique aspect. Equidistant mapping (oblique Plate Carr´ ee projection), conformal mapping (oblique Mercator projection), equal area mapping (oblique Lambert cylindrical equal area projection).
12-21 Equidistant mapping (oblique Plate Carr´ ee projection), compare with Fig. 12.2
x
y
= R
A cos B 0
B
,
Λ 1 =
cos B 0
cos B
, Λ 2 = 1 .
(12.4)
12-1 General mapping equations
Setting up general equations of the mapping “sphere to cylinder”: projections in the oblique aspect.
Meta-longitude, meta-latitude.
The general equations for mapping the sphere to a cylinder in the transverse aspect are based on the
general equation (10.1) of Chapter 10, but spherical longitude Λ and spherical latitude Φ being replaced
by their counterparts meta-longitude and meta-latitude, which are indicated here by capital letters
A and B. In order to treat simultaneously both the transverse tangent cylinder and the transverse
secant cylinder, we introduce B 0 as the meta-latitude of the meta-parallel circles B = ±B 0 which shall
be mapped equidistantly. In consequence, the general equations for this case are given by the very
general vector relation (12.1), taking into account the constraints (3.51) and (3.53), namely (12.2).
For the distortion analysis, the left principal stretches result to (12.3).
x
y
=
RA cos B 0
f (B)
,
(12.1)
tan A =
cos Φ sin(Λ − Λ 0 )
cos Φ sin Φ 0 cos(Λ − Λ 0 ) − sin Φ cos Φ 0
,
sin B = cos Φ cos Φ 0 cos(Λ − Λ 0 ) + sin Φ sin Φ 0 ,
(12.2)
Λ 1 =
cos B 0
cos B
, Λ 2 =
f
(B)
R
.
(12.3)
The procedure of how to set up special equations of the mapping sphere to cylinder in the oblique
aspect (oblique equidistant mapping, oblique conformal mapping, oblique equal area mapping)
can be easily deduced from the preceding chapters. Far easier, in the mapping equations as well
in the equations for the left principal stretches defined in Chapter 10, conventional coordinates
spherical longitude Λ and spherical latitudes Φ and Φ 0 are simply replaced by their corresponding
items meta-spherical longitude A and meta-spherical latitudes B and B 0 . Transformations of conventional spherical coordinates to meta-spherical coordinates is performed using (12.2).
12-2 Special mapping equations
Setting up special equations of the mapping “sphere to cylinder”: meta-cylindrical projections in the
oblique aspect. Equidistant mapping (oblique Plate Carr´ ee projection), conformal mapping (oblique Mercator projection), equal area mapping (oblique Lambert cylindrical equal area projection).
12-21 Equidistant mapping (oblique Plate Carr´ ee projection), compare with Fig. 12.2
x
y
= R
A cos B 0
B
,
Λ 1 =
cos B 0
cos B
, Λ 2 = 1 .
(12.4)