Surfaces of Gaussian Curvature Zero - Map Projections: Cartographic Information Systems - page 178

⎡ ⎤

R cos Φ 0 cos U − V sin Φ 0 cos U

⎣ ⎦ ,

X ( U, V )=[ E 1 , E 2 , E 3 ] R cos Φ 0 sin U V sin Φ 0 sin U

R sin Φ 0 + V cos Φ 0

−

⎡ ⎤

R cos Φ 0 cos U
R cos Φ 0 sin U
R sin Φ 0

⎣ ⎦ ∈ S

1
R cos Φ 0

x ( U )=[ E 1 , E 2 , E 3 ] , (4.5)

⎡ ⎤

− sin Φ 0 cos U

⎣ ⎦ =: C Φ .

Y ( U )=[ E 1 , E 2 , E 3 ] − sin Φ 0 sin U

cos Φ 0

Matrix of the metric:

2 + V 2 2 ,

G 11 = R 2 cos 2 Φ 0 + 1;

2 = D U X ( U, V )

G 11 = E = D U x D U Y

(4.6)

G 12 = G 21 = F = D U x ( U ) | Y ( U ) = D U X ( U, V ) |D V X ( U, V ) ,
G 12 = G 21 = 0; (4.7)

2 = 2 , G 22 = 1;

G 22 = G = Y ( U ) D V X ( U, V ) (4.8)

G= 1+ R 2 cos 2 Φ 0 0

.

(4.9)

0 1

Definition 4.1 (Ruled surface.).

2 of
type X ( U, V )= x ( U )+ V Y ( U ), where x ( U ) is a differentiable curve and Y ( U ) is a vector field
along the curve x ( U ) which vanishes nowhere.

A surface is called ruled surface if there exists a parameterization of the continuity class C

End of Definition.

The matrix of the metric, G, associated with a ruled surface is a typical element of this kind.
Compare with Box 4.2 , where we have computed the matrix G.

Box 4.2 (The matrix of the metric of a ruled surface).

2 + V 2 2 = D U X ( U, V ) 2 ,

G 11 = E = D U x D U Y (4.10)

G 12 = G 21 = F = D U x ( U ) | Y ( U ) = D U X ( U, V ) | D V X ( U, V ) , (4.11)

2 = 2 ,

G 22 = G = Y ( U ) D V X ( U, V ) (4.12)