242 8 “Ellipsoid-of-revolution to tangential plane”
Box 8.8 (Perspective mapping equations, minimal distance, perspective center Λ 0 , Φ 0 , H 0 ).
South coordinates:
x
∗ = x
∗ (p) =
= H 0
−N cos Φ sin Φ 0 cos(Λ − Λ 0 ) + N (1 − E
2 ) sin Φ cos Φ 0 + N 0 E
2 sin Φ 0 cos Φ 0
N cos Φ cos Φ 0 cos(Λ − Λ 0 ) + N (1 − E 2 ) sin Φ sin Φ 0 − N 0 + N 0 E 2 sin
2 Φ 0 − H 0
.
(8.76)
East coordinates:
y
∗ = y
∗ (p) =
= H 0
−N cos Φ sin(Λ − Λ 0 )
N cos Φ cos Φ 0 cos(Λ − Λ 0 ) + N (1 − E 2 ) sin Φ sin Φ 0 − N 0 + N 0 E 2 sin
2 Φ 0 − H 0
.
(8.77)
N :=
A 1
p
1 − E 2 sin
2 Φ
, N 0 :=
A 1
p
1 − E 2 sin
2 Φ 0
.
(8.78)
Alternatives.
East coordinates:
x
∗∗ = y
∗ .
(8.79)
North coordinates:
y
∗∗ = −x
∗ .
(8.80)
Polar coordinates (South azimuth α, radial coordinate r):
tan α
∗ =
=
N cos Φ sin(Λ − Λ 0 )
N cos Φ sin Φ 0 cos(Λ − Λ 0 ) − N (1 − E 2 ) sin Φ cos Φ 0 − N 0 E 2 sin Φ 0 cos Φ 0
,
r =
p
x ∗ 2 + y ∗ 2 =
p
x ∗∗ 2 + y ∗∗ 2 .
(8.81)
Alternative coordinates:
α
∗∗ = 90
◦ − α
∗
(East azimuth) .
(8.82)
At this point, you may enjoy our examples. Our first example, see Fig. 8.9, uses a tilted perspective,
also called Space Photo projection: the eastern seaboard viewed from a point about 160 km above
Newburgh, New York (Φ 0 = 41
◦ 30
northern latitude, Λ 0 = 74
◦ Western longitude, 1
◦ graticule). Our
second example, see Fig. 8.10, uses a tilted perspective, also called Space Photo projection: France and
Central Europe viewed from a point about 640 km above central Spain (Φ 0 = 40
◦ northern latitude,
Λ 0 = 5
◦ western longitude, 2
◦ graticule).