4-1 Ruled surfaces 155
Box 4.1 (Circular cone C
2
R cos Φ 0 of the sphere S
2
R : ruled surface).
Spherical coordinates and Cartan’s frame of reference:
X (Λ, Φ) =
ˆ
E 1 , E 2 , E 3
˜
2
4
R cos Φ cos Λ
R cos Φ sin Λ
R sin Φ
3
5 ,
(4.1)
C Λ :=
D Λ X (Λ, Φ)
D Λ X (Λ, Φ)
=
ˆ
E 1 , E 2 , E 3
˜
2
4
− sin Λ
+ cos Λ
0
3
5 ,
C Φ :=
D Φ X(Λ, Φ)
D Φ X(Λ, Φ)
=
ˆ
E 1 , E 2 , E 3
˜
2
4
− sin Φ cos Λ
− sin Φ sin Λ
cos Φ
3
5 .
(4.2)
Directrix x(U ) (parallel circle, small circle, curve of constant latitude Φ 0 , U = Λ):
x(U ) :=
ˆ
E 1 , E 2 , E 3
˜
2
4
R cos Φ 0 cos U
R cos Φ 0 sin U
R sin Φ 0
3
5 .
(4.3)
Generator or ruler of the surface (U = Λ):
Y (U ) := C Φ (Φ 0 ) =
ˆ
E 1 , E 2 , E 3
˜
2
4
− sin Φ 0 cos U
− sin Φ 0 sin U
cos Φ 0
3
5 .
(4.4)
Parameterized ruled surface:
X (U, V ) = x(U ) + V Y (U ) ,
X (U, V ) =
ˆ
E 1 , E 2 , E 3
˜
2
4
R cos Φ 0 cos U − V sin Φ 0 cos U
R cos Φ 0 sin U − V sin Φ 0 sin U
R sin Φ 0 + V cos Φ 0
3
5 ,
x(U ) =
ˆ
E 1 , E 2 , E 3
˜
2
4
R cos Φ 0 cos U
R cos Φ 0 sin U
R sin Φ 0
3
5 ∈ S
1
R cos Φ 0 ,
Y (U ) =
ˆ
E 1 , E 2 , E 3
˜
2
4
− sin Φ 0 cos U
− sin Φ 0 sin U
cos Φ 0
3
5 =: C Φ .
(4.5)
Matrix of the metric:
G 11 = E = D U x
2 +V
2 D U Y
2 = D U X (U, V )
2 , G 11 = R
2 cos
2 Φ 0 + 1 ;
(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)
G 22 = G = Y (U )
2 = D V X(U, V )
2 , G 22 = 1 ;
(4.8)
G =
»
1 + R
2 cos
2 Φ 0 0
0
1
–
.
(4.9)