Map Projections of Alternative Structures: Torus, Hyperboloid, Paraboloid, Onion Shape and Others - Map Projections: Cartographic Information Systems - page 637

Table 23.4 Inversion of a univariate homogeneous polynomial of degree n , clothoidal application

y( x )= a 11 x + a 12 x 2 + a 13 x 3 + a 14 x 4 + + a 1 n x n

···

x ( y )= b 11 y+ b 12 y 2 + b 13 y 3 + b 14 y 4 + + b 1 n y n

···

b 11 = a − 1

11

a − 3
11

b 12 = − a 12

a 12 a − 1
22

a 13 a − 1
33

b 13 = a − 1
11

a 23 −

a 12 a − 1 a 24 −

a 34 + a 13 a − 1 a 14 a − 1 b 14 = a − 1 a 23 a − 1 a 34 −

11 22 33 33 44

...

y:= x

− x 0 ,x := s − s 0

:( 1.17 ):

1
2 κ 0 sin α 0 ,

a 11 =cos α 0 ,a 12 = −

6 cos α 0 κ 0 + κ 0 tan α 0 ,a 14 = 24 κ 0 cos α 0 κ 0 tan α 0 −

3 κ 0 ,...

1 1

a 13 = −

r 0

: κ 0 =1 /r 0 ,κ 0 =0).

Example 23.1 (the circle S

α ( s )= α 0 + s s 0 )

x − x 0 = s 0 cos α 0 ( s ) ds = s 0 cos [ α 0 + κ 0 ( s − s 0 )] ds = κ 0 α 0 + κ 0 ( s−s 0 )

κ ( s ) ds = α 0 + κ 0 ( s −

s 0

cos xdx

y 0 = s
s 0

sin α 0 ( s ) ds = s
s 0

s 0 )] ds = κ 0 α 0 + κ 0 ( s−s 0 )

α 0

y − sin [ α 0 + κ 0 ( s − sin xdx

α 0

A series expansion of ( 23.146 ), ( 23.147 )accordingtoTable 23.1 leads directly to ( 23.139 ) with
κ 0 = 0. Similarly a series expansion of the root in ( 23.148 ) approaches ( 5.23 )for κ 0 =0.

End of Example.

r 0 2 ).

Corollary 23.3 (embedding of S in R

x 0 = sin [ α 0 + κ 0 ( s

− s 0 )] − sin α 0

x − (23.146)

κ 0

y 0 = −

cos [ α 0 + κ 0 ( s − s 0 )] + cos α 0

y − (23.147)

κ 0

End of Corollary.

s 0 )] = 1 s 0 )] 2 sin 2 [ α 0 + κ 0 ( s

cos [ α 0 + κ 0 ( s − − −

= 1 x 0 )+sin α 0 ] 2 − [ κ 0 ( x −