Ellipsoid-of-Revolution to Tangential Plane - Map Projections: Cartographic Information Systems - page 266

(tan( Δ/ 2)) = 1
2

1
cos 2 ( Δ/ 2) =

1

1+cos Δ , (sin Δ ) =cos Δ,

tan( Δ/ 2)
sin Δ

1
1+cos Δ cos Δ = 1

1

lim
Δ→ 0

= lim
Δ→ 0

2 ,

(8.56)

1+ E
1

E/ 2

√ 1 − E 2 =1

c
2 A 1

Δ→ 0 Λ 1 ( Δ )=

lim

− E

⇒

E 2 1

E/ 2

2 A 1
√ 1

E
1+ E

−

c = .

−

Parameterized conformal mapping:
α = Λ, r = f ( Δ ) ,

1 E/ 2 1+ E cos Δ

1

B/ 2

2 A 1
√ 1 − E 2

E
1+ E

−

tan Δ f ( Δ )=

2 ,

− E cos Δ

f ( Δ ) → f ( Φ ) , (8.57)

E 2 1 − E

E/ 2 1+ E sin Φ
1

E/ 2

tan π

,

2 A 1 Φ

2

f ( Φ )= √ 1

4 − 1+ E − E sin Φ

−

x = r cos α = f ( Φ )cos Λ, y = r sin α = f ( Φ )sin Λ.

Left principal stretches and left eigenvectors:
Λ 1 = Λ 2 = f ( Φ ) 1

E 2 sin 2 Φ
A 1 cos Φ

−

, (8.58)

D 4 X D Φ X

E Λ =

(“Easting”) , E Φ = = − E Δ (“Northing”) ,

D 4 X D Φ X

(8.59)

(i) C 1 Λ 1 = E Λ f ( Φ ) 1 , (ii) C 2 Λ 2 = E Φ f ( Φ ) 1

E 2 sin 2 Φ
A 1 cos Φ

E 2 sin 2 Φ
A 1 cos Φ

− −

.

Left angular shear:
Σ l = Ψ l − Ψ r =0 , Ω l =0 . (8.60)

Parameterized inverse mapping:
f ( x )= 1+ x
1 − x

E/ 2

= f (0) + 1

1! f ( x )

| x =0 ,

E cos Δ = x 1 , (8.61)