“Ellipsoid-of-Revolution to Cylinder”: Polar Aspect - Map Projections: Cartographic Information Systems - page 346

F 2 ( Φ )+ G 2 ( Φ )d Φ,

f ( Φ )
F 2 ( Φ )+ G 2 ( Φ ) =1

F (0)
F ( Φ )

d f = F ( Φ )
F (0)

Λ 1 Λ 2 =1 ⇒ ⇔

Φ

F ( Φ ) F 2 ( Φ )+ G 2 ( Φ )d Φ +const .,

1
F (0)

f (0) = 0 ⇒ f ( Φ )= (14.54)

0

f (0) = 0 ⇒ const . =0 .

The following formulae define the general mapping equations and the left principal stretches for
an equiareal cylindric mapping.

=

, Λ 1 = F (0)

x
y

F (0) Λ

F (0) 0 F ( Φ ) F 2 ( Φ )+ G 2 ( Φ )d Φ

F ( Φ ) ,

1

Λ 2 = F ( Φ )

1
Λ 1 .

F (0) =

(14.55)

14-34 An Example (Mapping the Torus)

A 1 B , especially for the parameter range A>B, 0 <U< 2 π ,

The torus is the product manifold S × S

and 0 <V < 2 π .

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

X
Y
Z

( A + B cos V )cos U
( A + B cos V )sin U
B sin V

( A + B cos Φ )cos Λ
( A + B cos Φ )sin Λ
B sin Φ

⎣ ⎦ = ⎣ ⎦ = ⎣ ⎦ .

(14.56)

The torus is the special surface which is generated by rotating a circle of radius B relative to a
circle of radius A>B around the center of the circle.

F ( Φ )= A + B cos Φ, G ( Φ )= B sin Φ,
U = Λ = arctan Y

Z
√ X 2 + Y 2

X , V = Φ = arctan

A .

(14.57)

−

Let us summarize in Box 14.3 the (left) tangent vectors and the (left) coordinates of the metric
tensor of the torus.

Box 14.3 (A special rotational figure: the torus).

Left tangent vectors:

G Λ := ∂ X

∂Λ =

− E 1 ( A + B cos Φ )sin Λ + E 2 ( A + B cos Φ )cos Λ, (14.58)

G Φ := ∂ X

∂Φ =

− E 1 B sin Φ cos Λ − E 2 B sin Φ sin Λ + E 3 B cos Φ,