Map Projections of Alternative Structures: Torus, Hyperboloid, Paraboloid, Onion Shape and Others - Map Projections: Cartographic Information Systems - page 616

2 onto a circular cylinder 2

Box 23.21 (Mapping paraboloid P C

1 : (i) conformal, (ii) equiareal,

(iii) equidistant, left principal stretches Λ 1 ,Λ 2 ).

(i) conformal:

x = U
y=1

√ 5+ln 1+ √ 5 + √ 1+4 V 2 +ln

V

1+ √ 1+4 V 2

−

Λ 1 = Λ 2 = 1
V

(ii) equiareal:

x = U

√ 1+4 V 2

12 + 1+4 V 2

1

y= −

12

Λ 1 = 1

V ,Λ 2 = V

(iii) equidistant:

x = U
y= 1

2 V √ 1+4 V 2 + 1 4 sinh − 1 2 V

Λ 1 = 1

V , Λ 2 =1

2 onto a circular cone 2

Box 23.22 (Mapping the paraboloid P C

0 :(i) conformal, (ii) equiareal,

(iii) equidistant, left principal stretches Λ 1 ,Λ 2 ).

(i) conformal:

x =( 1+ √ 5

1+ √ 1+4 V 2 exp √ 1+4 V 2 ) n cos ( nU )

V

exp √ 5

y =( 1+ √ 5

1+ √ 1+4 V 2 exp √ 1+4 V 2 ) n sin ( nU )

V

exp √ 5

1+ √ 5
exp √ 5 1+ √ 1+4 V 2 exp √ 1+4 V 2 n

Λ 1 = Λ 2 = n
V

V