188 5 “Sphere to tangential plane”: polar (normal) aspect
Box 5.15 (Gnomonic projection).
Parameterized central perspective mapping (polar coordinates):
α = Λ ,
r = R tan ∆ = R cot Φ .
(5.84)
Parameterized central perspective mapping (Cartesian coordinates):
"
x
y
#
= R cot Φ
"
cos Λ
sin Λ
#
.
(5.85)
Left principal stretches:
Λ 1 =
1
sin Φ
,
Λ 2 =
1
sin
2 Φ
.
(5.86)
Left eigenvectors:
C 1 Λ 1 = E Λ
1
sin Φ
(Easting) ,
C 2 Λ 2 = E Φ
1
sin
2 Φ
(Northing) .
(5.87)
Left maximal angular distortion:
Ω l = 2 arcsin
˛
˛
˛
˛
Λ 1 − Λ 2
Λ 1 + Λ 2
˛
˛
˛
˛ = 2 arcsin
˛
˛
˛
˛
1 − sin Φ
1 + sin Φ
˛
˛
˛
˛ .
(5.88)
Parameterized inverse mapping:
tan Λ =
y
x
, tan Φ =
R
p
x 2 + y 2
.
(5.89)
Note that the northern gnomonic projection covers all points in the half open interval π/2 ≤ Φ < 0.
The point Φ = 0 moves to infinity. Accordingly, for a complete gnomonic atlas of the sphere, we need
three charts: one northern, one southern, and one equatorial chart.
Question.
Question: “What made the gnomonic projection particularly useful in marine, aerial, and
space navigation?” Answer: “It is the property that the gnomonic projection is geodesic.
Geodesics, namely great circles of the sphere, are mapped onto a straight line – a very
important characteristic!”