6-2 Special mapping equations 211
Box 6.1 (Transverse equidistant mapping of the sphere to a plane at the meta-North Pole. Parameters:
Λ 0 ∈ [0
◦ , 360
◦ ], Φ 0 = 0).
Parameterized mapping:
α = A , r = f (B) = Rarc(π/2 − B) ,
(6.4)
x = r cos α = R (π/2 − B) cos A , y = r sin α = R (π/2 − B) sin A ,
tan A =
sin(Λ − Λ 0 )
− tan Φ
, sin B = cos Φ cos(Λ − Λ 0 ) .
(6.5)
Left principal stretches:
Λ 1 =
π/2 − B
cos B
, Λ 2 = 1 .
(6.6)
Left eigenvectors:
C 1 Λ 1 = E A
π/2 − B
sin (π/2 − B)
, C 2 Λ 2 = E B .
(6.7)
Parameterized inverse mapping:
tan A =
y
x
, B =
π
2
−
r
x 2 + y 2
R 2
,
tan(Λ − Λ 0 ) =
sin A
tan B
,
sin Φ = − cos B cos A .
(6.8)
Left maximum angular distortion:
Ω l = 2 arcsin
˛
˛
˛
˛
Λ 1 − Λ 2
Λ 1 + Λ 2
˛
˛
˛
˛ = 2 arcsin
˛
˛
˛
˛
π
2
− B − cos B
π
2
− B + cos B
˛
˛
˛
˛ .
(6.9)
6-22 Conformal mapping (transverse stereographic projection, transverse UPS)
The transverse conformal mapping of the sphere to a tangential plane is easily derived with the
knowledge of the preceding paragraph. We here conveniently rewrite the mapping equations of the
normal conformal mapping of Section 5-22 in terms of the (meta-)coordinates meta-longitude A and
meta-latitude B. Again, we take into account the relations (6.1) and (6.2) between meta-coordinates,
standard spherical coordinates (Λ, Φ), and the coordinates Λ 0 ∈ [0
◦ , 360
◦ ] and Φ 0 = 0
◦ of the metaNorth Pole. Then, the setup of the mapping equations is given by Lemma 6.1.