Map Projections of Alternative Structures: Torus, Hyperboloid, Paraboloid, Onion Shape and Others - Map Projections: Cartographic Information Systems - page 633

x − x 0 = s
s 0

y − y 0 = s
s 0

cos α ( s ) ds, sin α ( s ) ds

(23.138)

1 1

6 cos α 0 ( κ 0 + κ 0 tan α 0 ) Δs 3 +

Δx := x − x 0 = cos α 0 Δs −

2 κ 0 sin α 0 Δs 2

−

120 cosα 0 κ 0 +6 κ 0 κ

3 κ 0 Δs 5

1

3 κ 0 ) Δs 4 +

1

24 κ 0 cosα 0 ( κ 0 tan α 0 −

0 tan α 0 − −

(23.139)

720 κ 0 cosα 0 κ 0 tan α 0 −

15 κ 0 tan α 0 Δs 6 + O ( Δs 7 )

1

10 κ 0 κ 0 −

1

2 κ 0 cos α 0 Δs 2

1

6 cos α 0 ( κ 0 tan α 0 −

κ 0 ) Δs 3

Δy := y − y 0 = sin α 0 Δs − − −

120 cosα 0 κ 0 tan

3 κ 0 tan α 0 Δs 5 +

24 κ 0 cosα 0 ( κ 0 +3 κ 0 tan α 0 ) Δs 4 +

1 1

6 κ 0 κ 0 −

− (23.140)

720 κ 0 cosα 0 κ 0 +10 κ 0 κ 0 tan α 0

15 κ 0 Δs 6 + O ( Δs 7 )

1

−

accurate up to order seven in Δs := s − s 0 . The initial curvatures k 0 = k ( s 0 ) are classified as

1 )
(ii) κ 0 =1 /r 0 ( r 0 radius of circle

(i) κ 0 = 0 (straight line L

1
r 0

)
(iii) κ 0 = s 0 /a 2 ( a 2 positive parameter of clothoid

S

C a 2 )

End of Lemma.
For the proof we depart from the integrals

x 0 = s
s 0

x − cos α ( s ) ds

cos α 0 + κ 0 ( s s 0 ) 2 d ( s = s

s 0

s 0 )+ 1

2 κ 0 ( s

− − − s 0 ) (23.141)

y − y 0 = s
s 0

sin α ( s ) ds

= s
s 0

sin α 0 + κ 0 ( s s 0 ) 2 d ( s

s 0 )+ 1

2 κ 0 ( s

− − − s 0 ) , (23.142)

subject to

s 0 )+ 1

s 0 ) 2

2 κ 0 ( s

α ( s )= α 0 + Δα, Δα := κ 0 ( s − − (23.143)

Table 23.1 Power series of (cos( α 0 + Δα ) , sin( α 0 + Δα ))

cos( α 0 + Δα )=
cos α 0 − 1!

− 2! cos α 0 Δα 2 + 3! sin α 0 Δα 3 + sin α 0 Δα

sin α 0 Δα 5 + O c Δα 6
sin( α 0 + Δα )=sin α 0 − 1!

+ 4! cos α 0 Δα 4 − 5!

− 2! sin α 0 Δα 2 − 3! cos α 0 Δα 3

cos α 0 Δα

cos α 0 Δα 5 + O c Δα 6 + 4! sin α 0 Δα 4 + 5! +

By means of the uniformly convergent power series of type cos( α 0 + Δα )andsin( α 0 + Δα )
given in Table 23.1 and the powers Δα n =( α 0 + Δα ) n given in Table 23.2 up to n =5 we are
able to compute the fundamental integrals of Fresnel type outlined in Table 23.3 . Indeed due to