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{{short description|Surface containing a line through every point}}
[[File:Bez-regelfl0.svg|upright=1.25|thumb|Definition of a ruled surface: every point lies on a line]]

In [[geometry]], a [[Differential geometry of surfaces|surface]] {{mvar|S}} in [[3-dimensional Euclidean space]] is '''ruled''' (also called a '''scroll''') if through every [[Point (geometry)|point]] of {{mvar|S}}, there is a [[straight line]] that lies on {{mvar|S}}. Examples include the [[plane (mathematics)|plane]], the lateral surface of a [[cylinder (geometry)|cylinder]] or [[cone (geometry)|cone]], a [[conical surface]] with [[ellipse|elliptical]] [[directrix (rational normal scroll)|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 {{harv|Fuchs|Tabachnikov|2007}}.

The properties of being ruled or doubly ruled are preserved by [[projective map]]s, and therefore are concepts of [[projective geometry]]. In [[algebraic geometry]], ruled surfaces are sometimes considered to be surfaces in [[Affine space|affine]] or [[projective space]] over a [[Field (mathematics)|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 ==
[[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.

== Examples == 

=== Right circular cylinder ===
{{further|Right circular cylinder}}
[[File:Regelfl-zk.svg|thumb|upright=1.25|cylinder, cone]]

A right circular cylinder is given by the equation
:<math>x^2+y^2=a^2.</math>
It can be parameterized as
:<math>\mathbf x(u,v)=(a\cos u,a\sin u,v)</math>
:::<math>= (a\cos u,a\sin u,0) + v (0,0,1)</math>
:::<math>= (1-v) (a\cos u,a\sin u,0) + v (a\cos u,a\sin u,1).</math>
with 
:<math>\mathbf c(u) = (a\cos u,a\sin u,0),</math>
:<math>\mathbf r(u) = (0,0,1),</math>
:<math>\mathbf d(u) = (a\cos u,a\sin u,1).</math>

=== Right circular cone ===
{{further|Right circular cone}}

A right circular cylinder is given by the equation
:<math>x^2+y^2=z^2.</math>
It can be parameterized as
:<math>\mathbf x(u,v) = (\cos u,\sin u,1) + v (\cos u,\sin u,1)</math>
:::<math>= (1-v) (\cos u,\sin u,1) + v (2\cos u,2\sin u,2).</math>
with
:<math>\mathbf c(u) = (\cos u,\sin u,1),</math>
:<math>\mathbf r(u) = (\cos u,\sin u,1),</math>
:<math>\mathbf d(u) = (2\cos u,2\sin u,2).</math>
In this case one could have used the [[apex (geometry)|apex]] as the directrix, i.e.
:<math>\mathbf c(u) = (0,0,0)</math>
and
: <math>\mathbf r(u) = (\cos u,\sin u,1)</math>
as the line directions.

For any cone one can choose the apex as the directrix. This shows that ''the directrix of a ruled surface may degenerate to a point''.

=== Helicoid ===
{{further|Helicoid}}
[[File:Wendelfl-regelfl.svg|thumb|helicoid]]

A helicoid can be parameterized as
:<math>\mathbf x(u,v) = (v\cos u,v\sin u, ku)</math>
:::<math>= (0,0,ku) + v (\cos u, \sin u, 0)</math>
:::<math>= (1-v) (0,0,ku) + v (\cos u,\sin u, ku).</math>
The directrix
:<math>\mathbf c(u) = (0,0,ku)</math>
is the z-axis, the line directions are
:<math>\mathbf r(u) = (\cos u, \sin u, 0)</math>,
and the second directrix
:<math>\mathbf d(u) = (\cos u,\sin u, ku)</math>
is a [[helix]].

The helicoid is a special case of the [[Generalized helicoid#Ruled generalized helicoids|ruled generalized helicoids]].

=== Cylinder, cone and hyperboloids ===
{{further|Hyperboloid}}
[[File:Regelfl-phi-h.svg|upright=1.25|thumb|hyperboloid of one sheet for <math>\varphi=63^\circ</math>]]

The parametric representation 
:<math>\mathbf x(u,v) = (1-v) (\cos (u-\varphi), \sin(u-\varphi),-1) + v (\cos(u+\varphi), \sin(u+\varphi), 1)</math>
has two horizontal circles as directrices. The additional parameter <math>\varphi</math> allows to vary the parametric representations of the circles. For
:<math>\varphi=0</math> one gets the cylinder <math>x^2+y^2=1</math>,
:<math>\varphi=\pi/2</math> one gets the cone <math>x^2+y^2=z^2</math>,
:<math>0<\varphi<\pi/2</math> one gets a hyperboloid of one sheet with equation <math>\frac{x^2+y^2}{a^2}-\frac{z^2}{c^2}=1</math> and the semi axes <math>a=\cos\varphi, c=\cot\varphi</math>.

A hyperboloid of one sheet is a doubly ruled surface.

=== Hyperbolic paraboloid ===
{{further|Hyperbolic paraboloid}}
[[File:Hyp-paraboloid-ip.svg|upright=1.25|thumb|Hyperbolic paraboloid]]

If the two directrices in '''(CD)''' are the lines 
:<math>\mathbf c(u) =(1-u)\mathbf a_1 + u\mathbf a_2, \quad \mathbf d(u)=(1-u)\mathbf b_1 + u\mathbf b_2</math>
one gets 
:<math>\mathbf x(u,v)=(1-v)\big((1-u)\mathbf a_1 + u\mathbf a_2\big) + v\big((1-u)\mathbf b_1 + u\mathbf b_2\big)</math>,
which is the hyperbolic paraboloid that interpolates the 4 points <math>\mathbf a_1, \mathbf a_2, \mathbf b_1, \mathbf b_2</math> bilinearly.<ref>G. Farin: ''Curves and Surfaces for Computer Aided Geometric Design'', Academic Press, 1990, {{ISBN|0-12-249051-7}}, p. 250</ref>

The surface is doubly ruled, because any point lies on two lines of the surface.

For the example shown in the diagram: 
:<math>\mathbf a_1=(0,0,0),\; \mathbf a_2=(1,0,0),\; \mathbf b_1=(0,1,0),\; \mathbf b_2=(1,1,1).</math>
The hyperbolic paraboloid has the equation <math>z=xy</math>.

=== Möbius strip ===
{{further|Möbius strip}}
[[File:Moebius-str.svg|thumb|Möbius strip]]

The ruled surface
:<math>\mathbf x(u,v) = \mathbf c(u) + v \mathbf r(u)</math> 
with
:<math>\mathbf c(u) = (\cos2u,\sin2u,0)</math> (circle as directrix),
:<math>\mathbf r(u) = ( \cos u \cos 2 u , \cos u \sin 2 u, \sin u ) \quad 0\le u< \pi,</math>
contains a Möbius strip.

The diagram shows the Möbius strip for <math>-0.3\le v \le 0.3</math>.

A simple calculation shows <math>\det(\mathbf \dot c(0), \mathbf \dot r(0), \mathbf r(0)) \ne 0</math> (see next section). Hence the given realization of a Möbius strip is ''not developable''. But there exist developable Möbius strips.<ref>W. Wunderlich: ''Über ein abwickelbares Möbiusband'', Monatshefte für Mathematik 66, 1962, S. 276-289.</ref>

=== Further examples ===
* [[Conoid]]
* [[Catalan surface]]
* [[Developable roller]]s ([[oloid]], [[sphericon]])
* [[Tangent developable]]

== Developable surfaces ==
{{further|Developable surface}}

For the determination of the normal vector at a point one needs the [[partial derivative]]s of the representation <math>\mathbf x(u,v) = \mathbf c(u) + v \mathbf r(u)</math>:
:<math>\mathbf x_u = \mathbf \dot c(u)+ v \mathbf \dot r(u)</math>,
:<math>\mathbf x_v= \mathbf r(u)</math>.
Hence the normal vector is
:<math>\mathbf n = \mathbf x_u \times \mathbf x_v = \mathbf \dot c\times \mathbf r + v( \mathbf \dot r \times \mathbf r).</math>
Since <math>\mathbf n \cdot \mathbf r = 0</math> (A mixed product with two equal vectors is always 0), <math>\mathbf r (u_0)</math> is a tangent vector at any point <math>\mathbf x(u_0,v)</math>. The tangent planes along this line are all the same, if <math>\mathbf \dot r \times \mathbf r</math> is a multiple of <math>\mathbf \dot c\times \mathbf r</math>. This is possible only if the three vectors <math>\mathbf \dot c, \mathbf \dot r, \mathbf r</math> lie in a plane, i.e. if they are linearly dependent. The linear dependency of three vectors can be checked using the determinant of these vectors:

:The tangent planes along the line <math>\mathbf x(u_0,v) = \mathbf c(u_0) + v \mathbf r(u_0)</math> are equal, if 
:: <math>\det(\mathbf \dot c(u_0), \mathbf \dot r(u_0), \mathbf r(u_0)) = 0</math>.

A smooth surface with zero [[Gaussian curvature]] is called ''developable into a plane'', or just ''developable''. The determinant condition can be used to prove the following statement:
:A ruled surface <math>\mathbf x(u,v) = \mathbf c(u) + v \mathbf r(u)</math> is developable if and only if 
::<math>\det(\mathbf \dot c, \mathbf \dot r, \mathbf r) = 0</math> 
:at every point.<ref>W. Kühnel: ''Differentialgeometrie'', p. 58–60</ref>

The generators of any ruled surface coalesce with one family of its asymptotic lines. For developable surfaces they also form one family of its [[Line of curvature|lines of curvature]]. It can be shown that any developable surface is a cone, a cylinder, or a surface formed by all tangents of a space curve.<ref>G. Farin: p. 380</ref>

[[File:Torse-ee.svg|upright=1.6|thumb|Developable connection of two ellipses and its development]]
The determinant condition for developable surfaces is used to determine numerically developable connections between space curves (directrices). The diagram shows a developable connection between two ellipses contained in different planes (one horizontal, the other vertical) and its development.<ref>[https://www2.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf E. Hartmann: ''Geometry and Algorithms for CAD'', lecture note, TU Darmstadt, p.&nbsp;113]</ref>

An impression of the usage of developable surfaces in ''Computer Aided Design'' ([[CAD]]) is given in ''Interactive design of developable surfaces''.<ref>[http://www.geometrie.tugraz.at/wallner/abw.pdf Tang, Bo, Wallner, Pottmann: ''Interactive design of developable surfaces'', ACM Trans. Graph. (MONTH 2015), DOI: 10.1145/2832906]</ref>

A ''historical'' survey on developable surfaces can be found in ''Developable Surfaces: Their History and Application''.<ref>[[Snezana Lawrence]]: [https://www.researchgate.net/publication/257314770_Developable_Surfaces_Their_History_and_Application ''Developable Surfaces: Their History and Application''], in  Nexus Network Journal 13(3) · October 2011, {{doi|10.1007/s00004-011-0087-z}}</ref>

==Ruled surfaces in algebraic geometry==
{{Main|Ruled variety}}
In [[algebraic geometry]], ruled surfaces were originally defined as [[projective surface]]s in [[projective space]] containing a straight line through any given point. This immediately implies that there is a projective line on the surface through any given point, and this condition is now often used as the definition of a ruled surface: ruled surfaces are defined to be abstract projective surfaces satisfying this condition that there is a projective line through any point. This is equivalent to saying that they are [[birational]] to the product of a curve and a projective line. Sometimes a ruled surface is defined to be one satisfying the stronger condition that it has a [[fibration]] over a curve with fibers that are projective lines. This excludes the projective plane, which has a projective line though every point but cannot be written as such a fibration.

Ruled surfaces appear in the [[Enriques classification]] of projective complex surfaces, because every algebraic surface of [[Kodaira dimension]] <math>-\infty</math> is a ruled surface (or a projective plane, if one uses the restrictive definition of ruled surface). 
Every minimal projective ruled surface other than the projective plane is the projective bundle of a 2-dimensional vector bundle over some curve. The ruled surfaces with base curve of genus 0 are the [[Hirzebruch surface]]s.

==Ruled surfaces in architecture==
Doubly ruled surfaces are the inspiration for curved [[hyperboloid structure]]s that can be built with a [[latticework]] of straight elements, namely:
* Hyperbolic paraboloids, such as [[saddle roof]]s.
* Hyperboloids of one sheet, such as [[cooling tower]]s and some [[waste container|trash bin]]s.

The [[RM-81 Agena]] [[rocket engine]] employed straight [[cooling channel]]s that were laid out in a ruled surface to form the throat of the [[nozzle]] section.
<gallery>
File:Didcot power station cooling tower zootalures.jpg|Cooling [[Hyperboloid structure|hyperbolic towers]] at [[Didcot Power Station]], UK; the surface can be doubly ruled.
File:Ciechanow water tower.jpg|Doubly ruled water tower with [[Toroid (geometry)|toroidal]] tank, by Jan Bogusławski in [[Ciechanów]], Poland
File:Kobe port tower11s3200.jpg|A hyperboloid [[Kobe Port Tower]], [[Kobe]], Japan, with a double ruling.
File:First Shukhov Tower Nizhny Novgorod 1896.jpg| Hyperboloid water tower, 1896 in [[Nizhny Novgorod]].
File:Shukhov tower shabolovka moscow 02.jpg|The [[gridshell]] of [[Shukhov Tower]] in Moscow, whose sections are doubly ruled. 
File:Cremona, torrazzo interno 02 scala a chiocciola.JPG|A ruled helicoid spiral staircase inside [[Cremona]]'s [[Torrazzo of Cremona|Torrazzo]].
File:Nagytotlak.JPG|Village church in Selo, Slovenia: both the roof (conical) and the wall (cylindrical) are ruled surfaces.
File:W-wa Ochota PKP-WKD.jpg|A [[paraboloid|hyperbolic paraboloid]] roof of [[Warszawa Ochota railway station]] in [[Warsaw]], Poland.
File:Aodai-nonla-crop.jpg|A ruled [[Pointed hat|conical hat]].
File:Corrugated-fibro-roofing.jpg|Corrugated roof tiles ruled by parallel lines in one direction, and [[sinusoidal]] in the perpendicular direction
File:US Navy 091022-N-2571C-042 Seabees use a long board to screed wet concrete.jpg|Construction of a planar surface by ruling ([[screed]]ing) concrete
</gallery>

== References ==

=== Notes ===
{{Reflist}}

=== Sources ===
* {{Citation | last=do Carmo | first=Manfredo P. |author-link=Manfredo do Carmo | title=Differential Geometry of Curves and Surfaces | publisher=Prentice-Hall | edition=1st | year=1976 | isbn=978-0132125895}}
* {{Citation | last1=Barth | first1=Wolf P. | last2=Hulek | first2=Klaus | last3=Peters | first3=Chris A.M. | last4=Van de Ven | first4=Antonius | title=Compact Complex Surfaces | publisher= Springer-Verlag, Berlin | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. | isbn=978-3-540-00832-3 |mr=2030225 | year=2004 | volume=4 | doi=10.1007/978-3-642-57739-0}} 
* {{Citation | last1=Beauville | first1=Arnaud | title=Complex algebraic surfaces | publisher=[[Cambridge University Press]] | edition=2nd | series=London Mathematical Society Student Texts | isbn=978-0-521-49510-3 |mr=1406314 | year=1996 | volume=34 | doi=10.1017/CBO9780511623936}}
* {{citation|first=W. L.|last= Edge |authorlink= William Edge (mathematician)
|title=The Theory of Ruled Surfaces
|url=https://archive.org/details/theoryofruledsur029537mbp
|via=[[Internet Archive]]
|publisher=Cambridge University Press|year= 1931}}. Review: ''[[Bulletin of the American Mathematical Society]]'' 37 (1931), 791-793, {{doi|10.1090/S0002-9904-1931-05248-4}}
* {{citation|title=Mathematical Omnibus: Thirty Lectures on Classic Mathematics|first1=D. |last1=Fuchs|first2=Serge|last2=Tabachnikov|author2-link=Sergei Tabachnikov|publisher=American Mathematical Society|year=2007|isbn=9780821843161|page=228|url=https://books.google.com/books?id=IiG9AwAAQBAJ&pg=PA228|contribution=16.5 There are no non-planar triply ruled surfaces}}.
* {{citation|title=Problems and Solutions in Mathematics, 3103|editor-first=Ta-tsien |editor-last=Li|year=2011|publisher=[[World Scientific Publishing Company]]|isbn=9789810234805 | edition=2nd | url=https://books.google.com/books?id=lrMOjbLhR2IC&pg=PA156}}.
* {{Citation | last1=Hilbert | first1=David | author1-link=David Hilbert | last2=Cohn-Vossen | first2=Stephan | author2-link=Stephan Cohn-Vossen | title=Geometry and the Imagination | publisher=Chelsea | location=New York | edition=2nd | isbn=978-0-8284-1087-8 | year=1952 }}.
* {{SpringerEOM |id=R/r082790 |title=Ruled surface |first=V.A. |last=Iskovskikh}}
* {{Citation |first=John |last=Sharp |title=D-Forms: surprising new 3-D forms from flat curved shapes|publisher=Tarquin|year= 2008|isbn=978-1-899618-87-3}}. Review: Séquin, Carlo H. (2009), ''Journal of Mathematics and the Arts'' 3: 229–230, {{doi|10.1080/17513470903332913}}

==External links==
* {{MathWorld |title=Ruled Surface |id=RuledSurface}}
* [http://math.arizona.edu/~models/Ruled_Surfaces Ruled surface pictures from the University of Arizona]
* [http://www.rhino3.de/design/modeling/developable/ Examples of developable surfaces on the Rhino3DE website]

[[Category:Surfaces]]
[[Category:Differential geometry]]
[[Category:Differential geometry of surfaces]]
[[Category:Complex surfaces]]
[[Category:Algebraic surfaces]]
[[Category:Geometric shapes]]
[[Category:Analytic geometry]]