Ruled surface: Difference between revisions

Latest revision as of 03:51, 6 November 2025

Definition of a ruled surface: every point lies on a line

In geometry, a surface Template:Mvar in 3-dimensional Euclidean space is ruled (also called a scroll) if through every point of Template:Mvar, there is a straight line that lies on Template:Mvar. 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 Script error: No such module "Footnotes"..

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

File:Bez-regelfl.svg

Ruled surface generated by two Bézier curves as directrices (red, green)

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 $ℝ^{3}$ is described by a parametric representation of the form $𝐱 (u, v) = 𝐜 (u) + v 𝐫 (u)$ for $u$ varying over an interval and $v$ ranging over the reals.Template:Sfn It is required that $𝐫 (u) \neq (0, 0, 0)$ , and both $𝐜$ and $𝐫$ should be differentiable.Template:Sfn

Any straight line $v \mapsto 𝐱 (u_{0}, v)$ with fixed parameter $u = u_{0}$ is called a generator. The vectors $𝐫 (u)$ describe the directions of the generators. The curve $u \mapsto 𝐜 (u)$ 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 $𝐱 (u, v) = (1 - v) 𝐜 (u) + v 𝐝 (u)$

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

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

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File:Regelfl-zk.svg

cylinder, cone

A right circular cylinder is given by the equation $x^{2} + y^{2} = a^{2} .$ It can be parameterized as $\begin{aligned} 𝐱 (u, v) & = (a \cos u, a \sin u, v) \\ = (a \cos u, a \sin u, 0) + v (0, 0, 1) \\ = (1 - v) (a \cos u, a \sin u, 0) + v (a \cos u, a \sin u, 1) . \end{aligned}$ with

$𝐜 (u) = (a \cos u, a \sin u, 0),$
$𝐫 (u) = (0, 0, 1),$
$𝐝 (u) = (a \cos u, a \sin u, 1) .$

Right circular cone

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A right circular cylinder is given by the equation $x^{2} + y^{2} = z^{2} .$ It can be parameterized as $\begin{aligned} 𝐱 (u, v) & = (\cos u, \sin u, 1) + v (\cos u, \sin u, 1) \\ = (1 - v) (\cos u, \sin u, 1) + v (2 \cos u, 2 \sin u, 2) . \end{aligned}$ with

$𝐜 (u) = (\cos u, \sin u, 1),$
$𝐫 (u) = (\cos u, \sin u, 1),$
$𝐝 (u) = (2 \cos u, 2 \sin u, 2) .$

In this case one could have used the apex as the directrix, i.e. $𝐜 (u) = (0, 0, 0)$ and $𝐫 (u) = (\cos u, \sin u, 1)$ 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

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File:Wendelfl-regelfl.svg

helicoid

A helicoid can be parameterized as $\begin{aligned} 𝐱 (u, v) & = (v \cos u, v \sin u, k u) \\ = (0, 0, k u) + v (\cos u, \sin u, 0) \\ = (1 - v) (0, 0, k u) + v (\cos u, \sin u, k u) . \end{aligned}$ The directrix $𝐜 (u) = (0, 0, k u)$ is the z-axis, the line directions are $𝐫 (u) = (\cos u, \sin u, 0),$ and the second directrix $𝐝 (u) = (\cos u, \sin u, k u)$ is a helix.

The helicoid is a special case of the ruled generalized helicoids.

Cylinder, cone and hyperboloids

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File:Regelfl-phi-h.svg

hyperboloid of one sheet for

φ = 6 3^{\circ}

The parametric representation $𝐱 (u, v) = (1 - v) (\cos (u - φ), \sin (u - φ), - 1) + v (\cos (u + φ), \sin (u + φ), 1)$ has two horizontal circles as directrices. The additional parameter $φ$ allows to vary the parametric representations of the circles. For

$φ = 0$ one gets the cylinder $x^{2} + y^{2} = 1$ ,
$φ = π / 2$ one gets the cone $x^{2} + y^{2} = z^{2}$ ,
$0 < φ < π / 2$ one gets a hyperboloid of one sheet with equation $\frac{x^{2} + y^{2}}{a^{2}} - \frac{z^{2}}{c^{2}} = 1$ and the semi axes $a = \cos φ, c = \cot φ$ .

A hyperboloid of one sheet is a doubly ruled surface.

Hyperbolic paraboloid

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File:Hyp-paraboloid-ip.svg

Hyperbolic paraboloid

If the two directrices in (CD) are the lines $𝐜 (u) = (1 - u) 𝐚_{1} + u 𝐚_{2}, 𝐝 (u) = (1 - u) 𝐛_{1} + u 𝐛_{2}$ one gets $𝐱 (u, v) = (1 - v) ((1 - u) 𝐚_{1} + u 𝐚_{2}) + v ((1 - u) 𝐛_{1} + u 𝐛_{2})$ , which is the hyperbolic paraboloid that interpolates the 4 points $𝐚_{1}, 𝐚_{2}, 𝐛_{1}, 𝐛_{2}$ bilinearly.^[1]

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

For the example shown in the diagram: $𝐚_{1} = (0, 0, 0), 𝐚_{2} = (1, 0, 0), 𝐛_{1} = (0, 1, 0), 𝐛_{2} = (1, 1, 1) .$ The hyperbolic paraboloid has the equation $z = x y$ .

Möbius strip

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File:Moebius-str.svg

Möbius strip

The ruled surface $𝐱 (u, v) = 𝐜 (u) + v 𝐫 (u)$ with

$𝐜 (u) = (\cos 2 u, \sin 2 u, 0)$ (circle as directrix),
$𝐫 (u) = (\cos u \cos 2 u, \cos u \sin 2 u, \sin u) 0 \leq u < π,$

contains a Möbius strip.

The diagram shows the Möbius strip for $- 0.3 \leq v \leq 0.3$ .

A simple calculation shows $\det (\dot{c} (0), \dot{r} (0), 𝐫 (0)) \neq 0$ (see next section). Hence the given realization of a Möbius strip is not developable. But there exist developable Möbius strips.^[2]

Further examples

Developable surfaces

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For the determination of the normal vector at a point one needs the partial derivatives of the representation $𝐱 (u, v) = 𝐜 (u) + v 𝐫 (u)$ : $𝐱_{u} = \dot{c} (u) + v \dot{r} (u),$ $𝐱_{v} = 𝐫 (u) .$ Hence the normal vector is $𝐧 = 𝐱_{u} \times 𝐱_{v} = \dot{c} \times 𝐫 + v (\dot{r} \times 𝐫) .$ Since $𝐧 \cdot 𝐫 = 0$ (A mixed product with two equal vectors is always 0), $𝐫 (u_{0})$ is a tangent vector at any point $𝐱 (u_{0}, v)$ . The tangent planes along this line are all the same, if $\dot{r} \times 𝐫$ is a multiple of $\dot{c} \times 𝐫$ . This is possible only if the three vectors $\dot{c}, \dot{r}, 𝐫$ 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:

Template:Block indent

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: Template:Block indent

The generators of any ruled surface coalesce with one family of its asymptotic lines. For developable surfaces they also form one family of its 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.^[3]

File:Torse-ee.svg

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.^[4]

An impression of the usage of developable surfaces in Computer Aided Design (CAD) is given in Interactive design of developable surfaces.^[5]

A historical survey on developable surfaces can be found in Developable Surfaces: Their History and Application.^[6]

Ruled surfaces in algebraic geometry

Script error: No such module "Labelled list hatnote". In algebraic geometry, ruled surfaces were originally defined as projective surfaces 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 $- \infty$ 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 surfaces.

Ruled surfaces in architecture

Doubly ruled surfaces are the inspiration for curved hyperboloid structures that can be built with a latticework of straight elements, namely:

Hyperbolic paraboloids, such as saddle roofs.
Hyperboloids of one sheet, such as cooling towers and some trash bins.

The RM-81 Agena rocket engine employed straight cooling channels that were laid out in a ruled surface to form the throat of the nozzle section.

Cooling hyperbolic towers at Didcot Power Station, UK; the surface can be doubly ruled.

Cooling hyperbolic towers at Didcot Power Station, UK; the surface can be doubly ruled.
Doubly ruled water tower with toroidal tank, by Jan Bogusławski in Ciechanów, Poland

Doubly ruled water tower with toroidal tank, by Jan Bogusławski in Ciechanów, Poland
A hyperboloid Kobe Port Tower, Kobe, Japan, with a double ruling.

A hyperboloid Kobe Port Tower, Kobe, Japan, with a double ruling.
Hyperboloid water tower, 1896 in Nizhny Novgorod.

Hyperboloid water tower, 1896 in Nizhny Novgorod.
The gridshell of Shukhov Tower in Moscow, whose sections are doubly ruled.

The gridshell of Shukhov Tower in Moscow, whose sections are doubly ruled.
A ruled helicoid spiral staircase inside Cremona's Torrazzo.

A ruled helicoid spiral staircase inside Cremona's Torrazzo.
Village church in Selo, Slovenia: both the roof (conical) and the wall (cylindrical) are ruled surfaces.

Village church in Selo, Slovenia: both the roof (conical) and the wall (cylindrical) are ruled surfaces.
A hyperbolic paraboloid roof of Warszawa Ochota railway station in Warsaw, Poland.

A hyperbolic paraboloid roof of Warszawa Ochota railway station in Warsaw, Poland.
A ruled conical hat.

A ruled conical hat.
Corrugated roof tiles ruled by parallel lines in one direction, and sinusoidal in the perpendicular direction

Corrugated roof tiles ruled by parallel lines in one direction, and sinusoidal in the perpendicular direction
Construction of a planar surface by ruling (screeding) concrete

Construction of a planar surface by ruling (screeding) concrete

References

Notes

↑ G. Farin: Curves and Surfaces for Computer Aided Geometric Design, Academic Press, 1990, Template:ISBN, p. 250
↑ W. Wunderlich: Über ein abwickelbares Möbiusband, Monatshefte für Mathematik 66, 1962, S. 276-289.
↑ G. Farin: p. 380
↑ E. Hartmann: Geometry and Algorithms for CAD, lecture note, TU Darmstadt, p. 113
↑ Tang, Bo, Wallner, Pottmann: Interactive design of developable surfaces, ACM Trans. Graph. (MONTH 2015), DOI: 10.1145/2832906
↑ Snezana Lawrence: Developable Surfaces: Their History and Application, in Nexus Network Journal 13(3) · October 2011, Script error: No such module "CS1 identifiers".

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Sources

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Script error: No such module "citation/CS1".. Review: Bulletin of the American Mathematical Society 37 (1931), 791-793, Script error: No such module "CS1 identifiers".
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Script error: No such module "citation/CS1".. Review: Séquin, Carlo H. (2009), Journal of Mathematics and the Arts 3: 229–230, Script error: No such module "CS1 identifiers".

External links

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Ruled surface pictures from the University of Arizona
Examples of developable surfaces on the Rhino3DE website

[1] G. Farin: Curves and Surfaces for Computer Aided Geometric Design, Academic Press, 1990, Template:ISBN, p. 250

[2] W. Wunderlich: Über ein abwickelbares Möbiusband, Monatshefte für Mathematik 66, 1962, S. 276-289.

[3] G. Farin: p. 380

[4] E. Hartmann: Geometry and Algorithms for CAD, lecture note, TU Darmstadt, p. 113

[5] Tang, Bo, Wallner, Pottmann: Interactive design of developable surfaces, ACM Trans. Graph. (MONTH 2015), DOI: 10.1145/2832906

[6] Snezana Lawrence: Developable Surfaces: Their History and Application, in Nexus Network Journal 13(3) · October 2011, Script error: No such module "CS1 identifiers".

[1]

[2]

[3]

[4]

[5]

[6]

@@ Line 13: / Line 13: @@
 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>
+<math display="block"> \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}}
@@ Line 19: / Line 19: @@
 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>
+<math display="block"> \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>
@@ Line 32: / Line 32: @@
 A right circular cylinder is given by the equation
-:<math>x^2+y^2=a^2.</math>
+<math display="block">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 display="block">\begin{align}
-:::<math>= (a\cos u,a\sin u,0) + v (0,0,1)</math>
+\mathbf x(u,v) &= (a\cos u,a\sin u,v) \\
-:::<math>= (1-v) (a\cos u,a\sin u,0) + v (a\cos u,a\sin u,1).</math>
+&= (a\cos u,a\sin u,0) + v (0,0,1) \\
+&= (1-v) (a\cos u,a\sin u,0) + v (a\cos u,a\sin u,1).
+\end{align}</math>
 with
-:<math>\mathbf c(u) = (a\cos u,a\sin u,0),</math>
+* <math>\mathbf c(u) = (a\cos u,a\sin u,0),</math>
-:<math>\mathbf r(u) = (0,0,1),</math>
+* <math>\mathbf r(u) = (0,0,1),</math>
-:<math>\mathbf d(u) = (a\cos u,a\sin u,1).</math>
+* <math>\mathbf d(u) = (a\cos u,a\sin u,1).</math>
 === Right circular cone ===
@@ Line 46: / Line 48: @@
 A right circular cylinder is given by the equation
-:<math>x^2+y^2=z^2.</math>
+<math display="block">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 display="block">\begin{align}
-:::<math>= (1-v) (\cos u,\sin u,1) + v (2\cos u,2\sin u,2).</math>
+\mathbf x(u,v) &= (\cos u,\sin u,1) + v (\cos u,\sin u,1) \\
+&= (1-v) (\cos u,\sin u,1) + v (2\cos u,2\sin u,2).
+\end{align}</math>
 with
-:<math>\mathbf c(u) = (\cos u,\sin u,1),</math>
+* <math>\mathbf c(u) = (\cos u,\sin u,1),</math>
-:<math>\mathbf r(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>
+* <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>
+<math display="block">\mathbf c(u) = (0,0,0)</math>
 and
-: <math>\mathbf r(u) = (\cos u,\sin u,1)</math>
+<math display="block">\mathbf r(u) = (\cos u,\sin u,1)</math>
 as the line directions.
@@ Line 67: / Line 71: @@
 A helicoid can be parameterized as
-:<math>\mathbf x(u,v) = (v\cos u,v\sin u, ku)</math>
+<math display="block">\begin{align}
-:::<math>= (0,0,ku) + v (\cos u, \sin u, 0)</math>
+\mathbf x(u,v) &= (v\cos u,v\sin u, ku) \\
-:::<math>= (1-v) (0,0,ku) + v (\cos u,\sin u, ku).</math>
+&= (0,0,ku) + v (\cos u, \sin u, 0) \\
+&= (1-v) (0,0,ku) + v (\cos u,\sin u, ku).
+\end{align}</math>
 The directrix
-:<math>\mathbf c(u) = (0,0,ku)</math>
+<math display="block">\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>,
+<math display="block">\mathbf r(u) = (\cos u, \sin u, 0),</math>
 and the second directrix
-:<math>\mathbf d(u) = (\cos u,\sin u, ku)</math>
+<math display="block">\mathbf d(u) = (\cos u,\sin u, ku)</math>
 is a [[helix]].
@@ Line 85: / Line 91: @@
 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>
+<math display="block">\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=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>\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>.
+* <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.
@@ Line 98: / Line 104: @@
 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>
+<math display="block">\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>,
+<math display="block">\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>
@@ Line 106: / Line 112: @@
 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>
+<math display="block">\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>.
@@ Line 114: / Line 120: @@
 The ruled surface
-:<math>\mathbf x(u,v) = \mathbf c(u) + v \mathbf r(u)</math>
+<math display="block">\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 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>
+* <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.
@@ Line 134: / Line 140: @@
 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 display="block">\mathbf x_u = \mathbf \dot c(u)+ v \mathbf \dot r(u),</math>
-:<math>\mathbf x_v= \mathbf r(u)</math>.
+<math display="block">\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>
+<math display="block">\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
+{{block indent | em = 1.5 | text = 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 display="block">\det(\mathbf \dot c(u_0), \mathbf \dot r(u_0), \mathbf r(u_0)) = 0.</math>}}
-:: <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
+{{block indent | em = 1.5 | text = A ruled surface <math>\mathbf x(u,v) = \mathbf c(u) + v \mathbf r(u)</math> is developable if and only if <math display="block">\det(\mathbf \dot c, \mathbf \dot r, \mathbf r) = 0</math> at every point.<ref>W. Kühnel: ''Differentialgeometrie'', p. 58–60</ref>}}
-::<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>
@@ Line 190: / Line 193: @@
 === Sources ===
+{{refbegin}}
 * {{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}}
@@ Line 203: / Line 207: @@
 * {{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}}
+{{refend}}
 ==External links==

Ruled surface: Difference between revisions

Latest revision as of 03:51, 6 November 2025

Contents

Definition and parametric representation

Examples

Right circular cylinder

Right circular cone

Helicoid

Cylinder, cone and hyperboloids

Hyperbolic paraboloid

Möbius strip

Further examples

Developable surfaces

Ruled surfaces in algebraic geometry

Ruled surfaces in architecture

References

Notes

Sources

External links

Navigation menu

Ruled surface: Difference between revisions

Latest revision as of 03:51, 6 November 2025

Definition and parametric representation

Examples

Right circular cylinder

Right circular cone

Helicoid

Cylinder, cone and hyperboloids

Hyperbolic paraboloid

Möbius strip

Further examples

Developable surfaces

Ruled surfaces in algebraic geometry

Ruled surfaces in architecture

References

Notes

Sources

External links

Navigation menu

Search