286 11 “Sphere to cylinder”: transverse aspect
11-1 General mapping equations
Setting up general equations of the mapping “sphere to cylinder”: projections in the transverse aspect.
Meta-spherical longitude, meta-spherical 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 the transverse tangent cylinder and the transverse
secant cylinder, we introduce B 0 as the meta-latitude of those 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 (11.1), taking into account the constraints (3.51) and (3.53) for Φ 0 = 0
◦ ,
namely (11.2). For the distortion analysis, the left principal stretches result to (11.3).
x
y
=
RA cos B 0
f (B)
,
(11.1)
tan A =
sin(Λ − Λ 0 )
− tan Φ
, sin B = cos Φ cos(Λ − Λ 0 ) ,
(11.2)
Λ 1 =
cos B 0
cos B
, Λ 2 =
f
(B)
R
.
(11.3)
The procedure of how to set up special equations of the mapping “sphere to cylinder” in the transverse aspect (transverse equidistant mapping, transverse conformal mapping, transverse 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 by using (11.2).
11-2 Special mapping equations
Setting up special equations of the mapping “sphere to cylinder”: meta-cylindrical projections in the transverse aspect. Equidistant mapping (transverse Plate Carr´ ee projection), conformal mapping (transverse
Mercator projection), equal area mapping (transverse Lambert cylindrical equal area projection).
11-21 Equidistant mapping (transverse Plate Carr´ ee projection), see Fig. 11.2
x
y
= R
A cos B 0
B
,
Λ 1 =
cos B 0
cos B
, Λ 2 = 1 .
(11.4)
11-1 General mapping equations
Setting up general equations of the mapping “sphere to cylinder”: projections in the transverse aspect.
Meta-spherical longitude, meta-spherical 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 the transverse tangent cylinder and the transverse
secant cylinder, we introduce B 0 as the meta-latitude of those 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 (11.1), taking into account the constraints (3.51) and (3.53) for Φ 0 = 0
◦ ,
namely (11.2). For the distortion analysis, the left principal stretches result to (11.3).
x
y
=
RA cos B 0
f (B)
,
(11.1)
tan A =
sin(Λ − Λ 0 )
− tan Φ
, sin B = cos Φ cos(Λ − Λ 0 ) ,
(11.2)
Λ 1 =
cos B 0
cos B
, Λ 2 =
f
(B)
R
.
(11.3)
The procedure of how to set up special equations of the mapping “sphere to cylinder” in the transverse aspect (transverse equidistant mapping, transverse conformal mapping, transverse 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 by using (11.2).
11-2 Special mapping equations
Setting up special equations of the mapping “sphere to cylinder”: meta-cylindrical projections in the transverse aspect. Equidistant mapping (transverse Plate Carr´ ee projection), conformal mapping (transverse
Mercator projection), equal area mapping (transverse Lambert cylindrical equal area projection).
11-21 Equidistant mapping (transverse Plate Carr´ ee projection), see Fig. 11.2
x
y
= R
A cos B 0
B
,
Λ 1 =
cos B 0
cos B
, Λ 2 = 1 .
(11.4)