Ellipsoid-of-Revolution to Tangential Plane - Map Projections: Cartographic Information Systems - page 264

Parameterized equidistant mapping:
x = A 1 E 0 π Φ cos Λ

2 −

−

A 1 ( E 2 sin 2 Φ + E 4 sin 4 Φ + E 6 sin 6 Φ + E 8 sin 8 Φ + E 10 sin 10 Φ +O(E 12 ))
cos Λ,

−

(8.47)

y = A 1 E 0 π Φ sin Λ

2 −

−

A 1 ( E 2 sin 2 Φ + E 4 sin 4 Φ + E 6 sin 6 Φ + E 8 sin 8 Φ + E 10 sin 10 Φ +O(E 12 ))
sin Λ.

−

Left principal stretches and left eigenvectors:
Λ 1 = f ( Δ ) √ 1

= f ( Φ ) 1

E 2 sin 2 Φ
A 1 cos Φ

E 2 cos 2 Δ
A 1 sin Δ

− −

,Λ 2 =1 , (8.48)

D Λ X
D Λ X

D Φ X
D Φ X

C 1 = E Λ = (“Easting”) ,C 2 = E Φ =

= − E Δ (“Northing”) , (8.49)

(i) C 1 Λ 1 = E Λ f ( Φ ) 1

E 2 sin 2 Φ
A 1 cos Φ

−

, (ii) C 2 Λ 2 = E Φ = − E Δ .

Left angular distortion:

= 2arcsin

f ( Φ ) 1

d l =2arcsin

E 2 sin 2 Φ

Λ 1 Λ 2

Λ 1 + Λ 2

− − − A 1 cos Φ

f ( Φ ) 1

. (8.50)

E 2 sin 2 Φ + A 1 cos Φ −

Parameterized inverse mapping ,Λ = α, tan Λ = y/x ( r = x 2 + y 2 ):
Φ = π

r
A 1 E 0 −

r
A 1 E 0 −

r
A 1 E 0 −

r
A 1 E 0 2 −

F 2 sin 2 F 4 sin 4 F 6 sin 6

r
A 1 E 0 −

− F 8 sin 8 (8.51)

r

A 1 E 0 +O( E 12 ) .

−F 10 sin 10

Following the procedure that is outlined in Box 8.6 , we are immediately able to generate the
conformal mapping equations.