Map Projections of Alternative Structures: Torus, Hyperboloid, Paraboloid, Onion Shape and Others - Map Projections: Cartographic Information Systems - page 627

f V =0 → c = 0 (23.117)

V

f ( V )= ( n +1) ( n +1) / 2
n n/ 2

cos 2 n− 1 V ∗ sin V ∗

V

( n +1) 2 cos 2 V ∗ sin 2 V ∗ + a 2 [ cos 2 V ∗ (n + 1)

n] 2 dV ∗ × − (23.118)

Box 23.27 (Equidistant mapping of the church tower of the onion shape M := Z onto the

circular cylinder C

2

g ( V ) ).
Λ 2 ( V )=1 ∀V ↔ f = g 2 ( V )+ h 2 ( V )

(23.119)

f ( V )= g 2 ( V )+ h 2 ( V ) dV + c (23.120)

f V =0 → c = 0 (23.121)

f ( V )= V
V

(n + 1) 2 cos 2 V ∗ sin 2 V ∗ + a 2 [ cos 2 V ∗ (n + 1)

n] 2 dV ∗ (23.122)

cos n− 1 V ∗ ×

−

Box 23.28 is a collection of the final mapping equations of a church tower of onion shape Z

2

onto a developed circular cylinder C

g ( V ) of type conformal, equiareal and equidistant.
Figures 23.27 , 23.28 ,and 23.29 are a visualization of various map projections where as a theme
a cardoid onto a church tower of onion shape has been mapped as an object. This contribution
is based on Grafarend and Syffus ( 1998b ).

2

Box 23.28 (Mapping the church tower of onion shape Z

2 onto developed circular cylinder

C

2

g ( V ) : (i) conformal, (ii) equiareal, (iii) equidistant, left principal stretches Λ 1 ,Λ 2 ).

(i) Conformal:

a
√ n + 1 (

n

n +1 ) n/ 2 U

x =

√ n n

(n + 1) n+1

y=

(n + 1) 2 cos 2 V ∗ sin 2 V ∗ +a 2 [ cos 2 V ∗ (n + 1) − n] 2
cos V ∗ sin V ∗

V

dV ∗

V

Λ 1 = Λ 2 = [n / (n + 1)] n/ 2

1
cos n V sin V

(n + 1)