In geometry, a surface S in 3-dimensional Euclidean space is ruled (also called a scroll) if through every point of S, there is a straight line that lies on S. Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the right conoid, the helicoid, and the tangent developable of a smooth curve in space. A ruled surface can be described as the set of points swept by a moving straight line. For example, a cone is formed by keeping one point of a line fixed whilst moving another point along a circle. A surface is doubly ruled if through every one of its points there are two distinct lines that lie on the surface. The hyperbolic paraboloid and the hyperboloid of one sheet are doubly ruled surfaces. The plane is the only surface which contains at least three distinct lines through each of its points (Fuchs & Tabachnikov 2007). The properties of being ruled or doubly ruled are preserved by projective maps, and therefore are concepts of projective geometry. In algebraic geometry, ruled surfaces are sometimes considered to be surfaces in affine or projective space over a field, but they are also sometimes considered as abstract algebraic surfaces without an embedding into affine or projective space, in which case "straight line" is understood to mean an affine or projective line.
Definition and parametric representation
A surface in 3-dimensional Euclidean space is called a ruled surface if it is the union of a differentiable one-parameter family of lines. Formally, a ruled surface is a surface in R 3 {\displaystyle \mathbb {R} ^{3}} that is described by a parametric representation of the form
x ( u , v ) = c ( u ) + v r ( u ) {\displaystyle \mathbf {x} (u,v)=\mathbf {c} (u)+v\mathbf {r} (u)}
for u {\displaystyle u} varying over an interval and v {\displaystyle v} ranging over the reals. It is required that r ( u ) ≠ ( 0 , 0 , 0 ) {\displaystyle \mathbf {r} (u)\neq (0,0,0)} , and both c {\displaystyle \mathbf {c} } and r {\displaystyle \mathbf {r} } should be differentiable. Any straight line v ↦ x ( u 0 , v ) {\displaystyle v\mapsto \mathbf {x} (u_{0},v)} with fixed parameter u = u 0 {\displaystyle u=u_{0}} is called a generator. The vectors r ( u ) {\displaystyle \mathbf {r} (u)} describe the directions of the generators. The curve u ↦ c ( u ) {\displaystyle u\mapsto \mathbf {c} (u)} is called the directrix of the representation. The directrix may collapse to a point (in case of a cone, see example below). The ruled surface above may alternatively be described by
x ( u , v ) = ( 1 − v ) c ( u ) + v d ( u ) {\displaystyle \mathbf {x} (u,v)=(1-v)\mathbf {c} (u)+v\mathbf {d} (u)}
with the second directrix d ( u ) = c ( u ) + r ( u ) {\displaystyle \mathbf {d} (u)=\mathbf {c} (u)+\mathbf {r} (u)} . To go back to the first description starting with two non intersecting curves c ( u ) , d ( u ) {\displaystyle \mathbf {c} (u),\mathbf {d} (u)} as directrices, set r ( u ) = d ( u ) − c ( u ) . {\displaystyle \mathbf {r} (u)=\mathbf {d} (u)-\mathbf {c} (u).}
The geometric shape of the directrices and generators are of course essential to the shape of the ruled surface they produce. However, the specific parametric representations of them also influence the shape of the ruled surface.
Examples
Right circular cylinder
A right circular cylinder is given by the equation
x 2 + y 2 = a 2 . {\displaystyle x^{2}+y^{2}=a^{2}.}
It can be parameterized as
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