Editing Ruled surface (section)

== Definition and parametric representation ==
[[File:Bez-regelfl.svg|upright=1.3|thumb|Ruled surface generated by two [[Bézier curve]]s as directrices (red, green)]]

A [[Differential geometry of surfaces|surface]] in [[3-dimensional Euclidean space]] is called a ''ruled surface'' if it is the [[Union (set theory)|union]] of a differentiable one-parameter family of lines.
Formally, a ruled surface is a surface in <math>\mathbb R^3</math> is described by a [[parametric representation]] of the form
: <math>\quad \mathbf x(u,v) = \mathbf c(u) + v \mathbf r(u)</math>
for <math>u</math> varying over an interval and <math>v</math> ranging over the reals.{{sfn|do Carmo|1976|p=188}} It is required that <math>\mathbf r(u) \neq (0,0,0)</math>, and both <math>\mathbf c</math> and <math>\mathbf r</math> should be differentiable.{{sfn|do Carmo|1976|p=188}}

Any straight line <math>v \mapsto \mathbf x(u_0,v)</math> with fixed parameter <math>u=u_0</math> is called a ''generator''. The vectors <math>\mathbf r(u)</math> describe the directions of the generators. The curve <math>u\mapsto \mathbf c(u)</math> 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
: <math>\quad \mathbf x(u,v) = (1-v) \mathbf c(u) + v \mathbf d(u)</math>

with the second directrix <math>\mathbf d(u)= \mathbf c(u) + \mathbf r(u)</math>. To go back to the first description starting with two non intersecting curves <math>\mathbf c(u), \mathbf d(u)</math> as directrices, set <math>\mathbf r(u)= \mathbf d(u) - \mathbf c(u).</math>

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.