
66 Computing in Geographic Information Systems
Now substituting the values from equation 4.6
hR
sin
π
2
+
β
2
+ α
=
P F
sin ∠P V F
(4.8)
⇒ h sin α = 2 sin
β
2
cos(
β
2
− α) (4.9)
⇒ h sin α = 2 sin
β
2
cos
β
2
cos α + 2 sin
2
β
2
sin α (4.10)
⇒ sin(h − 2 sin
2
β
2
) = cos α sin β (4.11)
⇒ tan α =
sin β
h − 2 sin
2
β
2
(4.12)
Now putting the value of tan α in the equation one can obtain
ρ =
hR sin β
h − 2 sin
2
β
2
=
hR sin β
h − 1 + cos β
(4.13)
⇒ ρ =
hR cos Φ
h − 1 + sin Φ
(4.14)
Now substituting the radial distance in the equation 4.14 one can obtain
x = ρ sin(λ − λ
x
) (4.15)
⇒ x = (
hR sin β
h − 1 + cos β
) sin(λ − λ
x
) (4.16)
y = ρ cos(λ − λ
x
) (4.17)
⇒ y = (
hR sin β
h − 1 + cos β
) cos(λ − λ
x
) (4.18)
4.4.1