Korn–Lichtenstein and d’Alembert–Euler Equations - Map Projections: Cartographic Information Systems - page 710

d 1 = d B

d Q ( Q 0 ) ,d 2 = 1

d r B
d Q r ( Q 0 ) .
The standard coecients are given by
d 1 =cos B 0 ,

d 2 B

d Q 2 ( Q 0 ) , ··· ,d r = 1

2! r !

1

2 cos 2 B 0 tan B 0 ,

d 2 = −

1

6 cos 3 B 0 [1

tan 2 B 0 ] , d 3 = − − (D.48)

1

24 cos 4 B 0 tan B 0 [5

tan 2 B 0 ] ,

d 4 = −

1

120 cos 5 B 0 [5

18 tan 2 B 0 +tan 4 B 0 ] . d 5 = −

Second choice.
The first or remove step is materialized by a Taylor series expansion of Q ( B ) around Q 0 ( B 0 )
and leads directly to
Q = artanh(sin B ) = ln tan π

,

4 + B

(D.49)

2

Q = Q 0 + ΔQ = Q 0 + q ∀ q := ΔQ ,

B = B 0 + ΔB = B 0 + b ∀ b := ΔB

⇒
q = c 1 b + c 2 b 2 + ∞ c r b r

(D.50)

r =3

∀

d r Q
d B r ( B 0 ) .
Consequently, the second or restore step is based upon standard series inversion ( D.51 )
according to ( D.52 )and( D.53 ).

d 2 Q
d B 2 ( B 0 ) ,

c 1 = d Q

d B ( B 0 ) ,c 2 = 1

,c r = 1

···

2! r !

b = d 1 q + d 2 q 2 + ∞ d r q r ,

(D.51)

r =3

cos B 0 ,c 2 = 1

1 tan B 0

c 1 =

cos B 0 ,

2

c 3 = 1
6

1 1

24

tan B 0

cos B 0 [1 + 2 tan 2 B 0 ] ,c 4 = cos B 0 [5 tan 2 B 0 ] ,

(D.52)

1
120

1

cos B 0 [5 + 28 tan 2 B 0 +24tan 4 B 0 ] ,

c 5 =

N→∞ N→∞

c r b r d r q r , q = ⇔ b =

r =1 r =1

d 1 = 1

c 2

c 1 ,d 3 = 2 c 2

c 3 5 c 2

c 1

+ 5 c 2 c 3

c 4

c 1 ,d 2 =

−

c 1 − c 1 ,d 4 =

−

c 1 − c 1 .

(D.53)