Ellipsoid-of-Revolution to Tangential Plane - Map Projections: Cartographic Information Systems - page 270

Integration constant:

+ c,

E 2
4 E

1

2 f 2 = A 1 1

−

ln 1+ E cos Δ
1

1 1

1+ E cos Δ

−

E cos Δ − E cos Δ +

(8.70)

− 1 −

E 2
4 E

c = A 1 1

−

f 2 ( Δ =0)=0

f ( Δ =0)=0 ⇔ ⇔

ln 1+ E .

1
1 − E −

1
1+ E

1 − E +

Parameterized mapping equations:

f ( Φ ) ,
f ( Δ )= A 1 √ 1

f ( Δ ) →

− E 2

1
1 − E 2 + 2 E ln 1+ E

1 cos Δ

1 − E 2 cos 2 Δ − 2 E ln 1+ E cos Δ

1

1 − E − 1 − E cos Δ ,

(8.71)

f ( Φ )= A 1 √ 1 − E 2

1
1 2 E ln 1+ E

1 sin Φ

1 − E 2 sin 2 Φ − 2 E ln 1+ E sin Φ

1

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

E 2 + E −

− 1 − 1 −

(8.72)

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

E 2 sin 2 Φ
A 1 cos Φ

− A 1 cos Φ

, Λ 2 =

f ( Φ ) 1 − E 2 sin 2 Φ

, (8.73)

D Λ X D Φ X

C 1 = E Λ = (“Easting”) ,C 2 = E Φ = (“Northing”) , (8.74)

D Λ X D Φ X

(i) C 1 Λ 1 = E Λ f ( Φ ) 1

E 2 sin 2 Φ
A 1 cos Φ

− A 1 cos Φ

f ( Φ ) 1

(ii) C 2 Λ 2 = E Φ .

E 2 sin 2 Φ −

Left maximal angular distortion:
Ω l =2arcsin

=2arcsin

Λ 1 − Λ 2
Λ 1 + Λ 2

Λ 1 − 1
Λ 1 +1

= (8.75)

=2arcsin

E 2 sin 2 Φ ) A 1 cos 2 Φ

f 2 ( Φ )(1 − E 2 f sin 2 Φ )+ A 1 cos 2 Φ

f 2 ( Φ )(1

− −

.