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