192 5 “Sphere to tangential plane”: polar (normal) aspect
Box 5.16 (Orthographic projection).
Parameterized orthographic mapping (polar coordinates):
α = Λ , r = R cos Φ .
(5.92)
Parameterized orthographic mapping (Cartesian coordinates):
"
x
y
#
= R cos Φ
"
cos Λ
sin Λ
#
=
"
X
Y
#
.
(5.93)
Left principal stretches:
Λ 1 = 1 , Λ 2 = sin Φ .
(5.94)
Left eigenvectors:
C 1 Λ 1 = E Λ (Easting) , C 2 Λ 2 = E Φ sin Φ (Northing) .
(5.95)
Left maximal angular distortion:
Ω l = 2 arcsin
˛
˛
˛
˛
Λ 1 − Λ 2
Λ 1 + Λ 2
˛
˛
˛
˛ = 2 arcsin
˛
˛
˛
˛
1 − sin Φ
1 + sin Φ
˛
˛
˛
˛ .
(5.96)
Parameterized inverse mapping:
tan Λ =
y
x
, cos Φ =
p
x 2 + y 2
R
.
(5.97)
Fig. 5.24. Special perspective map of the sphere S
2
R onto the tangent space T N S
2
R : orthographic projection,
Tissot ellipses, polar aspect, graticule 15
◦ , shorelines.