Ruled surface

Ruled surface: definition

In geometry , a surface is called a ruled surface if applies

• A straight line goes through every point on the surface and is entirely contained in the surface.

Simple examples are planes , cylinders , cones, and single-shell hyperboloids . In the case of a single-shell hyperboloid, two straight lines go through each point. However, the following applies: A ruled surface in which three straight lines pass through each point can only be one plane.

In the case of specific ruled surfaces, the generators are often limited to distances in order to avoid infinite expansion (e.g. with cylinders or cones) or self-penetrations (e.g. with rule screw surfaces ).

In the term ruled surface, rule - as in the tilting rule  - has the original meaning of the Latin regula (stick, ruler), which is still contained in the English rule or the French règle today.

Ruled surfaces are used in architecture as easily modelable surfaces. For example, a cooling tower is often in the form of a single-shell hyperboloid. In the metal industry , developable ruled surfaces such as B. cylinder and cone used. (The single-shell hyperboloid cannot be developed!) Developable surfaces have the practical advantage that they can be produced from their developments from sheet metal by winding them (see Development (Descriptive Geometry) ). In the geometric modeling , ruled surfaces are z. B. used to generate Coons areas .

Definition and parameter representation

Ruled surface created with two Bezier curves as guide curves (red, green)

definition

• A two-dimensional differentiable manifold is called a ruled surface if it is the union of a single-parameter family of lines . The straight lines of this family are called the generators of the ruled surface.

Parametric representation A ruled surface can be represented by a parametric representation of the shape

• (CR) ${\displaystyle \quad \mathbf {x} (u,v)={\color {red}\mathbf {c} (u)}+v\;{\color {blue}\mathbf {r} (u)}}$

describe. Every surface curve with a fixed parameter is a generating line (straight line) and the curve is the guide curve . The vectors describe the directional field of the generators. ${\displaystyle \;\mathbf {x} (u_{0},v)\;}$ ${\displaystyle u=u_{0}}$ ${\displaystyle \;\mathbf {c} (u)\;}$ ${\displaystyle \;\mathbf {r} (u)\;}$

The ruled surface described by the parameter representation * can also be described as a second guide curve using the curve : ${\displaystyle \;\mathbf {d} (u)=\mathbf {c} (u)+\mathbf {r} (u)\;}$

• (CD) ${\displaystyle \quad \mathbf {x} (u,v)=(1-v)\;{\color {red}\mathbf {c} (u)}+v\;{\color {green}\mathbf {d} (u)}\ .}$

Conversely, one can assume two non-intersecting curves as guide curves and thus obtain the representation of a ruled surface with the direction field ${\displaystyle \;\mathbf {r} (u)=\mathbf {d} (u)-\mathbf {c} (u)\ .}$

When generating a ruled surface with the help of two guide curves (or a guide curve and a directional field), not only the geometric shape of these curves is important, but the specific parameter representation has a significant influence on the shape of the ruled surface. See examples d)

The representation (CR) is advantageous for theoretical investigations (see below) , since the parameter only occurs in one term. ${\displaystyle v}$

Examples

Ruled surfaces: cylinder, cone

a) Vertical circular cylinder : ${\displaystyle \ x^{2}+y^{2}=a^{2}\ }$

${\displaystyle \mathbf {x} (u,v)=(a\cos u,a\sin u,v)^{T}}$

${\displaystyle ={\color {red}(a\cos u,a\sin u,0)^{T}}\;+\;v\;{\color {blue}(0,0,1)^{T}}}$

${\displaystyle =(1-v)\;{\color {red}(a\cos u,a\sin u,0)^{T}}\;+\;v\;{\color {green}(a\cos u,a\sin u,1)^{T}}\ .}$

Here is ${\displaystyle \quad \mathbf {c} (u)=(a\cos u,a\sin u,0)^{T}\ ,\quad \mathbf {r} (u)=(0,0,1)^{T}\ ,\quad \mathbf {d} (u)=(a\cos u,a\sin u,1)^{T}\ .}$

b) Vertical circular cone : ${\displaystyle \ x^{2}+y^{2}=z^{2}\ }$

${\displaystyle \mathbf {x} (u,v)=(\cos u,\sin u,1)^{T}\;+\;v\;(\cos u,\sin u,1)^{T}}$

${\displaystyle =(1-v)\;(\cos u,\sin u,1)^{T}\;+\;v\;(2\cos u,2\sin u,2)^{T}.}$

Here you could have chosen as the leading curve , i.e. the tip of the cone, and as the direction field . The tip can be selected as the guide curve for all cones. ${\displaystyle \quad \mathbf {c} (u)=(\cos u,\sin u,1)^{T}\;=\;\mathbf {r} (u)\ ,\quad \mathbf {d} (u)=(2\cos u,2\sin u,2)^{T}\ .}$
${\displaystyle \ \mathbf {c} (u)=(0,0,0)^{T}\ }$ ${\displaystyle \ \mathbf {r} (u)=(\cos u,\sin u,1)^{T}\ }$

Helical surface as a ruled surface

c) helical surface :

${\displaystyle \mathbf {x} (u,v)=\;(v\cos u,v\sin u,ku)^{T}\;}$

${\displaystyle =\;(0,0,ku)^{T}\;+\;v\;(\cos u,\sin u,0)^{T}\ }$

${\displaystyle =\;(1-v)\;(0,0,ku)^{T}\;+\;v\;(\cos u,\sin u,ku)^{T}\ .}$

The guide curve is the z-axis, the direction field and the second guide curve is a helix . ${\displaystyle \ \mathbf {c} (u)=(0,0,ku)^{T}\;}$ ${\displaystyle \;\mathbf {r} (u)=\ (\cos u,\sin u,0)^{T}\;}$ ${\displaystyle \ \mathbf {d} (u)=(\cos u,\sin u,ku)^{T}\ }$

Ruled surface: single-shell hyperboloid for ${\displaystyle \varphi =63^{\circ }}$

d) Cylinders, cones and hyperboloids:
the parametric representation

${\displaystyle \mathbf {x} (u,v)=(1-v)\;(\cos(u-\varphi )\;,\;\sin(u-\varphi )\;,\;-1)^{T}\;+\;v\;(\cos(u+\varphi )\;,\;\sin(u+\varphi )\;,\;1)^{T}}$

has two horizontal unit circles as guide curves. The additional parameter allows the parameter representation of the circles to be varied. For ${\displaystyle \varphi }$

${\displaystyle \varphi =0\ }$one obtains the cylinder for ${\displaystyle x^{2}+y^{2}=1}$

${\displaystyle \varphi =\pi /2\ }$one gets the cone and for ${\displaystyle x^{2}+y^{2}=z^{2}}$

${\displaystyle 0<\varphi <\pi /2\ }$one obtains a single-shell hyperboloid with the equation and the semiaxes . ${\displaystyle \ {\tfrac {x^{2}+y^{2}}{a^{2}}}-{\tfrac {z^{2}}{c^{2}}}=1\ }$ ${\displaystyle \ a=\cos \varphi \;,\;c=\cot \varphi }$

Hyperbolic paraboloid

e) Hyperbolic paraboloid :

If the guidelines in (CD) are the straight lines

${\displaystyle \mathbf {c} (u)=(1-u)\mathbf {a} _{1}+u\mathbf {a} _{2},\quad \mathbf {d} (u)=(1-u)\mathbf {b} _{1}+u\mathbf {b} _{2}}$

are obtained

${\displaystyle \mathbf {x} (u,v)=(1-v){\big (}(1-u)\mathbf {a} _{1}+u\mathbf {a} _{2}{\big )}\ +\ v{\big (}(1-u)\mathbf {b} _{1}+u\mathbf {b} _{2}{\big )}\ }$.

This is the hyperbolic paraboloid that bilinearly interpolates the 4 points . For the example the drawing is ${\displaystyle \ \mathbf {a} _{1},\;\mathbf {a} _{2},\;\mathbf {b} _{1},\;\mathbf {b} _{2}\ }$

${\displaystyle \ \mathbf {a} _{1}=(0,0,0)^{T},\;\mathbf {a} _{2}=(1,0,0)^{T},\;\mathbf {b} _{1}=(0,1,0)^{T},\;\mathbf {b} _{2}=(1,1,1)^{T}\ }$.

and the hyperbolic paraboloid has the equation . ${\displaystyle z=xy}$

Möbius strip

f) Möbius strip :

The ruled surface

${\displaystyle \mathbf {x} (u,v)=\mathbf {c} (u)+v\;\mathbf {r} (u)}$

With

${\displaystyle \mathbf {c} (u)=(\cos 2u,\sin 2u,0)^{T}\ }$ (the leading curve is a circle),

${\displaystyle \mathbf {r} (u)=(\cos u\cos 2u,\cos u\sin 2u,\sin u)^{T}\ ,\quad 0\leq u<\pi \ ,}$

contains a Möbius strip.

The drawing shows the Möbius strip for . ${\displaystyle -0.3\leq v\leq 0.3}$

It is easy to calculate that is (see next section). I.e. this realization of a Möbius strip can not be developed . However, there are also unwindable Möbius strips. ${\displaystyle \det(\mathbf {\dot {c}} (0)\;,\;\mathbf {\dot {r}} (0)\;,\;\mathbf {r} (0))\;\neq \;0\ }$

Tangent planes, developable surfaces

For the derivations required here it is always assumed that they also exist.

To calculate the normal vector at a point, one needs the partial derivatives of the representation  : ${\displaystyle \quad \mathbf {x} (u,v)=\mathbf {c} (u)+v\;\mathbf {r} (u)}$

${\displaystyle \mathbf {x} _{u}=\mathbf {\dot {c}} (u)+v\;\mathbf {\dot {r}} (u)\ }$ , ${\displaystyle \quad \mathbf {x} _{v}=\;\mathbf {r} (u)}$

• ${\displaystyle \mathbf {n} =\mathbf {x} _{u}\times \mathbf {x} _{u}=\mathbf {\dot {c}} \times \mathbf {r} +v(\mathbf {\dot {r}} \times \mathbf {r} )\ .}$

Since the scalar product is (A late product with two equal vectors is always 0!), There is a tangent vector in every point . The tangent planes along this straight line are identical if is a multiple of . This is only possible if the three vectors lie in one plane, i.e. H. are linearly dependent. The linear dependence of three vectors can be determined with the help of the determinants of these vectors: ${\displaystyle \mathbf {n} \cdot \mathbf {r} =0}$ ${\displaystyle \mathbf {r} (u_{0})}$ ${\displaystyle \mathbf {x} (u_{0},v)}$ ${\displaystyle \mathbf {\dot {r}} \times \mathbf {r} }$ ${\displaystyle \mathbf {\dot {c}} \times \mathbf {r} }$ ${\displaystyle \mathbf {\dot {c}} \;,\;\mathbf {\dot {r}} \;,\;\mathbf {r} \ }$

• The tangent planes along the straight line are the same if ${\displaystyle \mathbf {x} (u_{0},v)=\mathbf {c} (u_{0})+v\;\mathbf {r} (u_{0})}$

${\displaystyle \det(\mathbf {\dot {c}} (u_{0})\;,\;\mathbf {\dot {r}} (u_{0})\;,\;\mathbf {r} (u_{0}))\;=\;0\ .}$

A generative for which this applies is called torsal .

• A ruled surface is genu then into a plane unwound when for all points Gaussian curvature vanishes. This is the case if and only if ${\displaystyle \quad \mathbf {x} (u,v)=\mathbf {c} (u)+v\;\mathbf {r} (u)}$

${\displaystyle \det(\mathbf {\dot {c}} \;,\;\mathbf {\dot {r}} \;,\;\mathbf {r} )\;=\;0\quad }$

applies in every point, d. i.e., if every generator is a torsal. A developable area is therefore also called a torse .

Properties of a developable surface:

• The generators represent a family of asymptote lines . They are also a family of lines of curvature .
• A developable surface is either a (general) cylinder or a (general) cone or a tangent surface (surface that consists of the tangents of a space curve).

Application and history of developable areas

Connection gap between two ellipses and their development

The determinant condition for developable areas gives one the possibility to numerically determine a connecting gate between two given guide curves. The picture shows an example of an application: connection gap between two ellipses (one horizontal, the other vertical) and their development.

An insight into the use of developable surfaces in CAD can be found in Interactive design of developable surfaces

Developable Surfaces: Their History and Application gives a historical overview of developable surfaces

Further examples

1. The envelope of a single -parameter set of planes
2. Hyperbolic paraboloid
3. Oloid
4. Catalan surface
5. Conoid
6. Standard screw surfaces

Striction line or throat line

definition

In the case of a cylindrical ruled surface, all generatrices are parallel; H. all direction vectors are parallel and thus with two parallel straight lines all points of one straight line have the same distance to the other straight line. ${\displaystyle \mathbf {r} (u)}$ ${\displaystyle {\dot {\mathbf {r} }}(u)=\mathbf {0} \ .}$

In the case of a non- cylindrical ruled surface, neighboring generatrices are skewed and there is a point on one straight line that is at a minimum distance from the other straight line. In this case such a point is called a central point . The totality of the central points form a curve, the strict line or throat line or waistline . The latter designation very clearly describes the stricture line of a single-shell rotational hyperboloid (see below). ${\displaystyle {\dot {\mathbf {r} }}(u)\neq \mathbf {0} \ .}$

• The amount of the Gaussian curvature assumes a maximum in the central point of a generator.

A cylindrical surface has no central points and therefore no strict line, or to put it clearly: no waist. In the case of a (general) conical surface, the stricture line / waist degenerates into a point, the apex of the cone.

Parametric representation

In the following considerations it is assumed that the ruled surface

${\displaystyle \mathbf {x} (u,v)=\mathbf {c} (u)+v\;\mathbf {r} (u)}$

is not cylindrical and sufficiently differentiable, more precisely:

${\displaystyle {\dot {\mathbf {r} }}(u)\neq \mathbf {0} \quad }$and for simplicity is. ${\displaystyle \quad |\mathbf {r} (u)|=1\ }$

The last property has the benefit of being what greatly simplifies calculations. In concrete examples, this property is usually not initially met. But what can be corrected through normalization. ${\displaystyle \quad \mathbf {r} \cdot {\dot {\mathbf {r} }}=0\quad }$

Two adjacent generators

${\displaystyle \mathbf {x} (v_{1})=\mathbf {c} (u)+v_{1}\;\mathbf {r} (u)}$

${\displaystyle \mathbf {x} (v_{2})=\mathbf {c} (u+\Delta u)+v_{2}\;\mathbf {r} (u+\Delta u)}$

At the end of the deliberations then goes . Therefore the following linear approximations (replace the curve in the immediate vicinity by its tangent) are useful: ${\displaystyle \Delta u\to 0}$

${\displaystyle \mathbf {c} (u+\Delta u)\approx \mathbf {c} (u)+\Delta u\;{\dot {\mathbf {c} }}(u)}$

${\displaystyle \mathbf {r} (u+\Delta u)\approx \mathbf {r} (u)+\Delta u\;{\dot {\mathbf {r} }}(u)}$.

Distance square

The square of the distance between two points on the straight line

${\displaystyle \mathbf {l} _{1}(v_{1})=\mathbf {c} +v_{1}\;\mathbf {r} }$

${\displaystyle \mathbf {l} _{2}(v_{2})=\mathbf {c} +\Delta u\;{\dot {\mathbf {c} }}+v_{2}\;(\mathbf {r} +\Delta u\;{\dot {\mathbf {r} }})\quad }$ is

${\displaystyle D(v_{1},v_{2})={\Big (}(v_{2}-v_{1})\;\mathbf {r} +\Delta u({\dot {\mathbf {c} }}+v_{2}\;{\dot {\mathbf {r} }}){\Big )}^{2}\ .}$

Parameters of the central point

The distance becomes minimal when the function becomes minimal. And this is the case when the 1st partial derivatives are zero: ${\displaystyle D(v_{1},v_{2})}$

${\displaystyle D_{v_{1}}=-2{\Big (}(v_{2}-v_{1})\;\mathbf {r} +\Delta u({\dot {\mathbf {c} }}+v_{2}\;{\dot {\mathbf {r} }}){\Big )}\cdot \mathbf {r} }$

${\displaystyle \ =-2{\color {magenta}(v_{2}-v_{1}+\Delta u\;{\dot {\mathbf {c} }}\cdot \mathbf {r} )}=0\ ,}$

${\displaystyle D_{v_{2}}=2{\Big (}(v_{2}-v_{1})\;\mathbf {r} +\Delta u({\dot {\mathbf {c} }}+v_{2}\;{\dot {\mathbf {r} }}){\Big )}\cdot (\mathbf {r} +\Delta u\;{\dot {\mathbf {r} }})}$

${\displaystyle \ =2{\Big (}{\color {magenta}v_{2}-v_{1}+\Delta u\;{\dot {\mathbf {c} }}\cdot \mathbf {r} }+{\Delta u}^{2}({\dot {\mathbf {c} }}\cdot {\dot {\mathbf {r} }}+v_{2}{\dot {\mathbf {r} }}^{2}){\Big )}=0\ .}$

From this system of equations for follows for : ${\displaystyle v_{1},v_{2}}$ ${\displaystyle \Delta u\to 0\ }$

• ${\displaystyle \quad v_{1}=v_{2}=-{\frac {{\dot {\mathbf {c} }}\cdot {\dot {\mathbf {r} }}}{{\dot {\mathbf {r} }}^{2}}}\ .}$

Parametric representation

So the parametric representation of the stricture line is

• ${\displaystyle \mathbf {x} (u)=\mathbf {c} (u)-{\frac {{\dot {\mathbf {c} }}(u)\cdot {\dot {\mathbf {r} }}(u)}{{\dot {\mathbf {r} }}^{2}(u)}}\;\mathbf {r} (u)\ .}$

Double ruled surfaces

There are two sets of straight lines on both the single-shell hyperboloid and the hyperbolic paraboloid . A strict line belongs to each family. In the single-shell rotational hyperbolod, the two striction lines coincide.

Examples

1) Single- shell rotational hyperboloid

${\displaystyle \mathbf {x} (u,v)={\begin{pmatrix}\cos u\\\sin u\\0\end{pmatrix}}+v\cdot {\begin{pmatrix}-\sin u\\\cos u\\k\end{pmatrix}}\ ,\ }$

The central points all have the parameter , i.e. H. the strict line is the unit circle in the xy plane. ${\displaystyle v=0}$

Striction lines (red) of a single-shell rotational hyperboloid, hyperbolic paraboloid and helical surface

2) Straight conoid

In the case of a straight conoid, the axis is the common perpendicular of all generators. (The general rule is: A point pair of two skewed straight lines has the shortest distance if its connection is the common perpendicular of the straight lines.) So it applies to straight conoids

The axis of a straight conoid is also its stricture line.

Examples of straight conoids are the hyperbolic paraboloid and the helical surface . ${\displaystyle z=xy}$

Screw gate, purple: guide curve and strict line

3) Torse

Each developable ruled surface (torse) other than the general cylinder and cone is a tangent surface, i.e. H. the totality of the generatrices of the ruled surface consists of the family of tangents of a given curve . (In the picture, the curve is a helical line. This creates a screw gate .) The general rule is ${\displaystyle \gamma }$

The strict line of a tangent surface created by a curve is the curve itself. ${\displaystyle \gamma }$ ${\displaystyle \gamma }$

4) Möbius strip

Strict line (red) of a Moebius band

For the description given above of a Möbius strip

${\displaystyle \mathbf {c} (u)=(\cos 2u,\sin 2u,0)^{T}\ }$ ,

${\displaystyle \mathbf {r} (u)=(\cos u\cos 2u,\cos u\sin 2u,\sin u)^{T}\ .}$

(About the picture: The band has been widened so that the striction line lies completely on the area shown.) In this case, the direction vector is already a unit vector, which considerably simplifies the calculation. ${\displaystyle \mathbf {r} }$

For the parameter of the respective central point, there is finally the parameter representation of the strict line ${\displaystyle v={\frac {4\cos u}{1+4\cos ^{2}u}}}$

${\displaystyle \mathbf {x} (u)={\frac {1}{1+4\cos ^{2}u}}\;{\big (}\cos(2u),\sin(2u),-2\sin(2u){\big )}\ .}$

It is easy to see that this curve lies in the plane . To show that this even curve is even ${\displaystyle 2y+z=0}$

is an ellipse with a center point and the semi-axes , ${\displaystyle (-{\tfrac {2}{5}},0,0)}$ ${\displaystyle a=1,b={\tfrac {3}{5}}}$

one shows that the x and y coordinates satisfy the equation . So the outline of the strict line is an ellipse and thus the strict line as a parallel projection too. ${\displaystyle \ {\tfrac {(x+{\tfrac {2}{5}})^{2}}{({\tfrac {3}{5}})^{2}}}+{\tfrac {y^{2}}{\tfrac {1}{5}}}=1\ }$

The strict line can be made easier by the parametric representation

${\displaystyle \mathbf {x} (t)=\mathbf {f} _{0}+\mathbf {f} _{1}\cos t+\mathbf {f} _{2}\sin t\ }$ With

${\displaystyle \mathbf {f} _{0}=(-{\tfrac {2}{5}},0,0)^{T},\ \mathbf {f} _{1}=({\tfrac {3}{5}},0,0)^{T},\ \mathbf {f} _{2}=(0,-{\tfrac {1}{\sqrt {5}}},{\tfrac {2}{\sqrt {5}}})^{T}}$

describe (see ellipse ).

Composition of ruled surfaces

It can be two ruled developable surface along a straight line or cut off and put them together so that out and a common straight line of the composite surface is treated with a new common tangent plane thereof. ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle g}$ ${\displaystyle h}$

In the case of a non-developable and a developable ruled surface, the surface thus combined cannot be differentiated along the common generatrix . The common generatrix is ​​visible as an edge, whereby the edge emerges differently and clearly at different points of the generators. In the case of two non-developable ruled surfaces, the surface composed in this way can be differentiable along the common generating line, but in general it is not.

Extra mathematical application

Ruled surfaces can be used not only in mathematics, but also outside of it in design and engineering. A good example of this is the work of the architect / mathematician Antoni Gaudí . The vault of La Sagrada Família describes several hyperboloids, hyperbolic paraboloids and helicoids.

literature

• Manfredo P. do Carmo: Differential geometry of curves and surfaces . Springer-Verlag, 2013, ISBN 978-3-322-85494-0 , pp. 142,147
• G. Farin: Curves and Surfaces for Computer Aided Geometric Design . Academic Press, 1990, ISBN 0-12-249051-7
• D. Hilbert, S. Cohn-Vossen: Illustrative geometry . Springer-Verlag, 2013, ISBN 978-3-662-36685-1 , p. 181
• W. Kühnel: Differential Geometry . Vieweg, 2003, ISBN 3-528-17289-4
• H. Schmidbauer: Developable surfaces: A design theory for practitioners . Springer-Verlag, 2013, ISBN 978-3-642-47353-1

Individual evidence

1. ^ DB Fuks, Serge Tabachnikov: There are no non-planar triply ruled surfaces . In: Mathematical Omnibus: Thirty Lectures on Classic Mathematics . American Mathematical Society, 2007, ISBN 978-0-8218-4316-1 , p. 228.
2. ↑ rule. In: Jacob Grimm , Wilhelm Grimm (Hrsg.): German dictionary . tape 14 : R - skewness - (VIII). S. Hirzel, Leipzig 1893 ( woerterbuchnetz.de ).
3. ^ G. Farin: Curves and Surfaces for Computer Aided Geometric Design , Academic Press, 1990, ISBN 0-12-249051-7 , p. 250
4. ↑ W. Wunderlich: About a developable Möbius tape, monthly books for mathematics 66, 1962, pp. 276–289.
5. ↑ W. Kühnel: Differentialgeometrie , pp. 58-60
6. ↑ G. Farin: p. 380
7. ↑ CAD script. (PDF) p. 113
8. ↑ Tang, Bo, Wallner, Pottmann: Interactive design of developable surfaces (PDF) In: ACM Trans. Graph. , (MONTH 2015), doi: 10.1145 / 2832906
9. ↑ Snezana Lawrence: developable Surfaces: Their History and Application . In: Nexus Network Journal , 13 (3), October 2011, doi: 10.1007 / s00004-011-0087-z
10. ^ W. Kühnel: Differentialgeometrie , Vieweg, 2003, ISBN 3-528-17289-4 , p. 58.
11. ↑ MP do Carmo: Differential geometry of curves and surfaces , Springer-Verlag, 2013, ISBN 3322850722 , p. 145.
12. ^ W. Haack : Elementare Differentialgeometrie , Springer-Verlag, 2013, ISBN 3034869509 , p. 32
13. ↑ about Gaudi's secret . Southgerman newspaper
14. ↑ about regular surfaces in the "Sagrada Familia". Science blogs

Web links

• A. Niggas: Ruled surfaces - theoretical, exemplary, visual