236 8 “Ellipsoid-of-revolution to tangential plane”
Continuation of Box.
Parameterized conformal mapping:
α = Λ , r = f (∆) ,
f (∆) =
2A 1
√
1 − E 2
„
1 − E
1 + E
« E/2 „
1 + E cos ∆
1 − E cos ∆
« E/2
tan
∆
2
,
f (∆) → f (Φ) ,
f (Φ) =
2A 1
√
1 − E 2
„
1 − E
1 + E
« E/2 „
1 + E sin Φ
1 − E sin Φ
« E/2
tan
„
π
4
−
Φ
2
«
,
x = r cos α = f (Φ) cos Λ , y = r sin α = f (Φ) sin Λ .
(8.57)
Left principal stretches and left eigenvectors:
Λ 1 = Λ 2 =
f (Φ)
p
1 − E 2 sin
2 Φ
A 1 cos Φ
,
(8.58)
E Λ =
D Λ X
D Λ X
(“Easting”) , E Φ =
D Φ X
D Φ X
= −E ∆ (“Northing”) ,
(i) C 1 Λ 1 = E Λ
f (Φ)
p
1 − E 2 sin
2 Φ
A 1 cos Φ
, (ii) C 2 Λ 2 = E Φ
f (Φ)
p
1 − E 2 sin
2 Φ
A 1 cos Φ
.
(8.59)
Left angular shear:
P
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 ∆ = x 1 ,
f (x) =
„
1 + x
1 − x
« E/2
= 1 +
1
1!
E
2
„
1 + x
1 − x
« E/2−1 (1 − x) − (1 + x)(−1)
(1 − x) 2
˛
˛
˛
˛
˛
x=0
x + O(2) =
= 1 + Ex + O(2) .
(8.61)
Alternative:
artanhx = x +
x
3
3
+
x
5
5
+
x
7
7
+
x
9
9
+
x
11
11
+ O(x
13 ) ,
E cos ∆ = x 1 ,
artanh(E cos ∆) =
= E cos ∆ +
E
3
3
cos
3 ∆ +
E
5
5
cos
5 ∆ +
E
7
7
cos
7 ∆ +
E
9
9
cos
9 ∆ +
E
11
11
cos
11 ∆ + O(x
13 ) .
(8.62)