Map Projections of Alternative Structures: Torus, Hyperboloid, Paraboloid, Onion Shape and Others - Map Projections: Cartographic Information Systems - page 608

√ 1+ tanh 2 x
1 −tanh 2 x

=1+ 2 tanh 2 x + 1 8 tanh 4 x + 2 16 tanh 6 x + O 8 ( tanhx )
∀|

(23.37)

tanhx | < 1

tanh 2 xdx =

3
2

3

2 tanhx + 3

−

2 x

(23.38)

tanh 4 xdx =

11
8

11 11

8 tanhx + 11 24 tanh 3 x

− −

8 x

(23.39)

tanh 6 xdx =

23
16

23 23 23

16 tanhx + 23 80 tanh 5 x 48 tanh 3 x

− − −

16 x

(23.40)

1

tanh 2 x − 1 1+ tanh 2 xdx = x 1+ 3

2 + 11 8 + 23

− − tanh x ·

16

3

− tanh 3 x 11

2 + 11 8 + 23 24 + 23

23

80 tanh 5 x + O 7 ( tanhx )

−

(23.41)

16 48

2 := 2 onto the

Box 23.14 (Equidistant mapping of the rotational symmetric hyperboloid M H

2

circular cylinder C

1 ).

f = √ cosh 2 V = 1+ tan h 2 V/ 1 tanh 2 V Λ 2 ( V )=1 ∀ V → − (23.42)

f ( V )= 1+ tanh 2 V

1 − tanh 2 V

dV + c (23.43)

f (0) = 0 = c → (23.44)

1 1

3 tanh 3 V 10 tanh 5 V + O 7 ( tanhV )

f ( V )=3 V − 2 tanhV − − (23.45)

Box 23.15 (The integral √ 1+ tanh 2 x

√ 1 −tanh 2 x dx ).

x 2 − 1 / 2 √ 1+ x 2 =1+ x 2 + 1

1

2 x 4 + 1 2 x 6 + O 8 ( x )

− ∀| x | < 1 (23.46)

√ 1+ tanh 2 x

√ 1 −tanh 2 x =1+ tanh 2 x + 2 tanh 4 x + 2 tanh 6 x + O 8 (tanh x )
∀|

(23.47)

tanhx | < 1

tanh 2 xdx = − tanh x + x

(23.48)

tanh 4 xdx =

1
2

1 1

2 tanhx + 1 6 tanh 3 x

− −

2 x

(23.49)

tanh 6 xdx = −

1
2

1 1 1

2 tanhx + 1 10 tanh 5 x − 6 tanh 3 x −

2 x

(23.50)