RULED SURFACES WITH CONSTANT PARAMETER OF ...

210 E. Kasap, M. Masal
From (2.2), a ruled surface M with constant distribution parameter has the
following differential equation:
x2x ′ 3 − x3x ′ 2 + (1 − x21 )τ − x1x3κ
= C X ′
, φ(s) = 0,
where C is a constant. Hence, we have
x1 = x2κds + c1,
x3 = x2( C|| X ′ || 2
1 − x2 1
x 2 1 + x22 + x23 = 1.
2
− τ)ds + c2,
(2.3)
The second equation of the triple (2.3) is an integral equation for the unknown
x3 = x3(s). Therefore, if x2 = x2(s) is given, we obtain x1 = x1(s) and
x3 = x3(s). Thus, we have the following theorem.
Theorem 2.1. The range of existence of ruled surfaces M with constant
distribution parameter comprises within one arbitrary function of one variable.
From the equalities
λ X ′ + v X ′ ∧
X
2
= λ 2
X ′
and
X ′
∧ X
= X ′
,
2
+ v 2
X ′ ∧
X
the unit normal vectors to the ruled surface M at (s , v ) and (s , o) are
n(s , v ) = λ ℓ(s) + v ℓ(s) ∧ X(s)
√ v 2 + λ 2
References
2
, n(s , o) = ℓ(s) = X ′ (s)
X ′
.
(s)
[1] M. Chasles, Corresp. Mathem. et Phys. De Quetelet, 11 (1839).
[2] M.P. Docarmo, Differential Geometry of Curves and Surfaces, Prentice-
Hall, Englewood Cliffs (1976).
[3] W.L. Edge, The Theory of Ruled Surfaces, Cambridge Univ. Press., Cambridge,
UK (1931)