218 7 “Sphere to tangential plane”: oblique aspect
Fig. 7.3. Mapping the sphere to a tangential plane: oblique aspect, conformal mapping. Point-of-contact:
meta-North Pole at Rio de Janeiro (Λ 0 = −43
◦ 12
, Φ 0 = −22
◦ 54
).
7-23 Equal area mapping (oblique Lambert projection)
The oblique equal area mapping of the sphere to a tangential plane is the generalization of equations
derived earlier. The results are stated more precisely in Box 7.3. Figure 7.4 gives an impression of
the famous oblique conformal mapping of the sphere to a tangential plane with the meta-North Pole
located at Perth (Λ 0 = −115
◦ 52
, Φ 0 = −31
◦ 57
).
Box 7.3 (Oblique equal area mapping of the sphere to a plane at the meta-North Pole Λ 0 ∈ [0
◦ , 360
◦ ],
Φ 0 ∈ [−90
◦ , 90
◦ ]).
Parameterized mapping:
α = A , r = f (B) = 2R tan
„
π
4
−
B
2
«
, x = 2R sin
„
π
4
−
B
2
«
cos A , y = 2R sin
„
π
4
−
B
2
«
sin A ,
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 .
(7.14)
Left principal stretches:
Λ 1 =
1
cos
` π
4
−
B
2
´ , Λ 2 = cos
„
π
4
−
B
2
«
.
(7.15)
Left eigenvectors:
C 1 Λ 1 = E A
1
cos
` π
4
−
B
2
´ , C 2 Λ 2 = E B cos
„
π
4
−
B
2
«
.
(7.16)
Parameterized inverse mapping:
tan A =
y
x
, sin
„
π
4
−
B
2
«
=
1
2R
p
x 2 + y 2 ,
tan(Λ − Λ 0 ) =
sin A
tan B cos Φ 0 + cos A sin Φ 0
, sin Φ = − cos B cos A cos Φ 0 + sin B sin Φ 0 .
(7.17)
Left maximum angular distortion:
Ω l = 2 arcsin
˛
˛
˛
˛
Λ 1 − Λ 2
Λ 1 + Λ 2
˛
˛
˛
˛ = 2 arcsin
˛
˛
˛
˛
˛
1 − cos
2
` π
4
−
B
2
´
1 + cos 2
` π
4
−
B
2
´
˛
˛
˛
˛
˛
.
(7.18)