Surfaces of Gaussian Curvature Zero - Map Projections: Cartographic Information Systems - page 177

C 2 R cos Φ 0 , directrix: parallel circle S 1 R cos Φ 0 , generating vector field

Fig. 4.1. Ruled surface of type circular cone

C Φ ( Λ,Φ 0 )

Fig. 4.2. Helicoid as ruled surface, one-sheeted hyperboloid of revolution as ruled surface

⎡ ⎤

− sin Φ 0 cos U

⎣ ⎦ .

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

cos Φ 0

(4.4)

Parameterized ruled surface:

X ( U, V )= x ( U )+ V Y ( U ) ,