Map Projections of Alternative Structures: Torus, Hyperboloid, Paraboloid, Onion Shape and Others - Map Projections: Cartographic Information Systems - page 623

M 2 := Z 2 ; onion constants

Fig. 23.25. Oblique orthogonal projection of the church tower of onion shape

n =3 ,α = β = γ =20 ◦

Example : n =3:= V π
6

(ii) g ( V )= a sin Vcos n− 2 V n ( n cos 2 V ( n +1) 2 − 1) −

g V = a 1

cos 2 V cos 2 V ( n− 2) / 2 ( n +1) 2 cos 2 V ] − [ n ( n − 1) −

g V = a 1

n

n +1 ) ( n− 2) / 2 [ n ( n

→ − n/ ( n +1)( − 1) − n ( n +1)]

(iii) g V = acos n VsinV = a cos 2 V n/ 2 1

cos 2 V −

g V = a n
n +1

n/ 2

n/ ( n +1)= a n
n +1

n/ 2

1

( n +1) − 1 / 2 → −

g V =

a
√ n +1 (

n

n +1 ) n/ 2

+

→ ∀ n ∈ N ,a ∈ R (23.98)

Indeed g ( V )accordingto( 23.98 ) as the maximal radius of the onion function determines the
radius of the cylinder in contact with

2 . The mapping equations are generated in such a way
that along the metaequator V of the church tower of onion shape the mapping onto the circular
cylinder of contact is equidistant. Consult Fig. 23.26 for an illustration of the projection geometry.
We begin with a general set up o f the m apping equations for the case of a circular cylinder of

Z

radius g V =a[ n/ ( n +1)] n/ 2 / ( n + 1) s computed by ( 23.98 )ofabove,namely

= g V U
f ( V )

x
y

(23.99)

2 . Thus we are left with the problem to
determine the unknown function f ( V ) by following the constructive approach of the theory of

in terms of Cartesian coordinates ( x, y )whichcover R