Rule area – Wikipedia

In geometry, an area is called Control area If one goes straight through every point of the area that is contained in the area.

This applies, for example, to levels, cylinders, cones, single -minded hyperboloids and hyperbolic paraboloids. In the latter two, even go through each point two Straight (there are double-curved surfaces). A control area where every point more than two Go straight, can only be one level. ^[first]

In the case of regular surfaces with finite expansion (e.g. cylinders) and without self -penetration (e.g. for bowling and control screw surfaces), the generators are limited to routes.

Concepted the control area Rule – as in Kippregel – the original meaning of Latin rule (Stab, Lineal), ^[2] The still in English today rule Or French rule is included.

Regular areas are used in architecture as easily modelable areas, since despite the curvature they can be composed of straight components or – in the case of concrete – they can be switched on with straight boards. Large cooling towers, for example, often have the shape of a single -minded hyperboloid. When construction of ventilation channels and in the case of plumbing work, sheet metal handling is used, i.e. handling control surfaces such as cylinder and cone segments, as these can be shaped by simply bending without stretching or compressing the material (as with the more complex process of the solid formation).
See also processing (performing geometry)

In the case of geometric modeling, rule areas z. B. used to produce COONS areas.

Definition

Parameter presentation A control area can be used by a parameter display of the form

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

describe. Every area curve

x ( in 0 , in ) {displaystyle ;mathbf {x} (u_{0},v);}

with a fixed parameter

in = in 0 {Displaystyle u = u_ {0}}

is a generating (straight) and the curve

c ( in ) {displaystyle ;mathbf {c} (u);}

is the Leading curve . The vectors

r ( in ) {displaystyle ;mathbf {r} (u);}

Describe that Direction of direction of the producing.

The control area described by the parameter representation * can also be used with the help of the curve

d ( in ) = c ( in ) + r ( in ) {displaystyle ;mathbf {d} (u)=mathbf {c} (u)+mathbf {r} (u);}

describe as a second lead curve:

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

Conversely, you can assume two corners that are not cutting corners and thus receive the representation of a control area with the directional field

r ( in ) = d ( in ) – c ( in ) . {displaystyle ;mathbf {r} (u)=mathbf {d} (u)-mathbf {c} (u) .}

When generating a control area with the help of two leading curves (or a lead curve and a direction of direction), not only the geometric shape of these curves is important, but the concrete parameter representation has a significant influence on the shape of the control surface. See examples d)

For theoretical studies (see below), the presentation is (CR) advantageous because the parameter

in {displaystyle v}

only occurs in one term.

Vertical circular cylinder [ Edit | Edit the source text ]

x 2 + and 2 = a 2 {displaystyle x^{2}+y^{2}=a^{2} }

:

x ( in , in ) = ( a cos ⁡ in , a sin ⁡ in , in ) T{Displaystyle Mathbf {x} (u, v) = (harsh u, asin u, v)^{t}}}

=(acos⁡u,asin⁡u,0)T+v(0,0,1)T{displaystyle ={color {red}(acos u,asin u,0)^{T}};+;v;{color {blue}(0,0,1)^{T}}}

=(1−v)(acos⁡u,asin⁡u,0)T+v(acos⁡u,asin⁡u,1)T .{displaystyle =(1-v);{color {red}(acos u,asin u,0)^{T}};+;v;{color {green}(acos u,asin u,1)^{T}} .}

Here is

c ( in ) = ( a cos ⁡ in , a sin ⁡ in , 0 ) T , r ( in ) = ( 0 , 0 , first ) T , d ( in ) = ( a cos ⁡ in , a sin ⁡ in , first ) T . {displaystyle quad mathbf {c} (u)=(acos u,asin u,0)^{T} ,quad mathbf {r} (u)=(0,0,1)^{T} ,quad mathbf {d} (u)=(acos u,asin u,1)^{T} .}

Vertical district cone [ Edit | Edit the source text ]

x 2 + and 2 = With 2 {displaystyle x^{2}+y^{2}=z^{2} }

:

x ( in , in ) = ( cos ⁡ in , sin ⁡ in , first ) T+ in ( cos ⁡ in , sin ⁡ in , first ) T{displaystyle mathbf {x} (u,v)=(cos u,sin u,1)^{T};+;v;(cos u,sin u,1)^{T}}

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

Here is

c ( in ) = ( cos ⁡ in , sin ⁡ in , first ) T = r ( in ) , d ( in ) = ( 2 cos ⁡ in , 2 sin ⁡ in , 2 ) T . {displaystyle quad mathbf {c} (u)=(cos u,sin u,1)^{T};=;mathbf {r} (u) ,quad mathbf {d} (u)=(2cos u,2sin u,2)^{T} .}

One would also have as a leading curve

c ( in ) = ( 0 , 0 , 0 ) T {displaystyle mathbf {c} (u)=(0,0,0)^{T} }

, the tip of the cone, and as a field of direction

r ( in ) = ( cos ⁡ in , sin ⁡ in , first ) T {displaystyle mathbf {r} (u)=(cos u,sin u,1)^{T} }

can choose. With all bowling you can choose the tip as a leading curve.

Spiral area [ Edit | Edit the source text ]

x ( in , in ) = ( in cos ⁡ in , in sin ⁡ in , k in ) T{Displaystyle mathbf {x} (u, v) =; (vcos u, vsin u, ku)^{t};};

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

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

The lead curve

c ( in ) = ( 0 , 0 , k in ) T {displaystyle mathbf {c} (u)=(0,0,ku)^{T};}

is the z-axis, the direction of direction

r ( in ) = ( cos ⁡ in , sin ⁡ in , 0 ) T {displaystyle ;mathbf {r} (u)= (cos u,sin u,0)^{T};}

and the second lead curve

d ( in ) = ( cos ⁡ in , sin ⁡ in , k in ) T {displaystyle mathbf {d} (U) = (cos u, sin the, ku) ^ {t}

is a screw line.

Cylinder, cone and hyperboloid [ Edit | Edit the source text ]

The parameter representation

x ( in , in ) = ( first – in ) ( cos ⁡ ( in – Phi ) , sin ⁡ ( in – Phi ) , – first ) T+ in ( cos ⁡ ( in + Phi ) , sin ⁡ ( in + Phi ) , first ) T{displaystyle mathbf {x} (U, v) = (1-V); (cos (U-varphi);,; cos (u+varphi);,; sin (u+varphi);,; 1)^{t}}

has two horizontal unit circles as leading curves. The additional parameter

Phi {displaystyle varphi }

allows the parametric representations of the circles to vary. For

Phi = 0 {displaystyle varphi =0 }

you get the cylinder x 2+ and 2= first {displaystyle x^{2}+y^{2}=1}

, for

Phi = Pi / 2 {displaystyle varphi = pi /2}

you get the cone x 2+ and 2= With 2{displaystyle x^{2}+y^{2}=z^{2}}

and for

0 < Phi < Pi / 2 {Displaystyle 0

If you get a single -minded hyperboloid with the equation x2+y2a2– z2c2= first {displaystyle {tfrac {x^{2}+y^{2}}{a^{2}}}-{tfrac {z^{2}}{c^{2}}}=1 }

and the half axles a = cos ⁡ Phi , c = cot ⁡ Phi {displaystyle a = cos varphi;,; c = cot varphi}

.

Hyperbolisches Paraboloid [ Edit | Edit the source text ]

If the guidelines in (CD) The straight

c ( in ) = ( first – in ) a1+ in a2, d ( in ) = ( first – in ) b1+ in b2{displaystyle mathbf {c} (u)=(1-u)mathbf {a} _{1}+umathbf {a} _{2},quad mathbf {d} (u)=(1-u)mathbf {b} _{1}+umathbf {b} _{2}}

are, you get

x ( in , in ) = ( first – in ) (( first – in ) a1+ in a2) + in (( first – in ) b1+ in b2) {displaystyle mathbf {x} (u,v)=(1-v){big (}(1-u)mathbf {a} _{1}+umathbf {a} _{2}{big )} + v{big (}(1-u)mathbf {b} _{1}+umathbf {b} _{2}{big )} }

.

This is the hyperbolic paraboloid that the 4 points

a first , a 2 , b first , b 2 {displaystyle mathbf {a} _{1},;mathbf {a} _{2},;mathbf {b} _{1},;mathbf {b} _{2} }

bilinear interpoliert. ^[3] For the example of the drawing is

a1= ( 0 , 0 , 0 ) T, a2= ( first , 0 , 0 ) T, b1= ( 0 , first , 0 ) T, b2= ( first , first , first ) T {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

With = x and {Displaystyle z = xy}

.

Möbiusband [ Edit | Edit the source text ]

The control area

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

with

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

(The lead curve is a circle),

r ( in ) = ( cos ⁡ in cos ⁡ 2 in , cos ⁡ in sin ⁡ 2 in , sin ⁡ in ) T , 0 ≤ in < Pi , {displaystyle mathbf {r} (u)=(cos ucos 2u,cos usin 2u,sin u)^{T} ,quad 0leq u

Contains a möbius band.

The drawing shows the Möbiusband for

– 0.3 ≤ in ≤ 0.3 {displaystyle -0.3leq vleq 0.3}

.

It is easily calculated that

the ( c˙( 0 ) , r˙( 0 ) , r ( 0 ) ) ≠ 0 {Displaystyle det(mathbf {dot {c}} (0);,;mathbf {dot {r}} (0);,;mathbf {r} (0));neq ;0 }

is (see next section). D. h. This realization of a möbius band is unobstructable . However, there are also handicapped furniture bands. ^[4]

Further examples [ Edit | Edit the source text ]

1. The enveloping of a saved level
2. Oloid
3. Catalan area
4. Konoid
5. Control surfaces

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

In order to calculate the normal vector in one point, you need the partial derivations of the presentation

x ( in , in ) = c ( in ) + in r ( in ) {displaystyle quad mathbf {x} (u,v)=mathbf {c} (u)+v;mathbf {r} (u)}

:

xu= c˙( in ) + in r˙( in ) {Displaystyle Mathbf {x} _{U}=Mathbf {Dot {C}} (u)+V;Mathbf {Dot {R}} (u)

, xv= r ( in ) {displaystyle quad mathbf {x} _{v}=;mathbf {r} (u)}

• 
  n = xu× xu= c˙× r + in ( r˙× r ) {Displaystyle Mathbf {N} =Mathbf {x} _{U}Times Mathbf {x} _{U}=Mathbf {Dot {c}} Times Mathbf {R} +V(Mathbf {Dot {R}} Times Mathbf { r} )}
  .

Because the scalar product

n ⋅ r = 0 {displaystyle mathbf {n} cdot mathbf {r} =0}

is (a spatal product with two same vectors is always 0!), is

r ( in 0 ) {displaystyle mathbf {r} (u_{0})}

A tangent vector in every point

x ( in 0 , in ) {displaystyle mathbf {x} (u_{0},v)}

. The tangential levels along these are identical if

r˙× r {displaystyle mathbf {dot {r}} times mathbf {r} }

many times from

c˙× r {displaystyle mathbf {dot {c}} times mathbf {r} }

is. This is only possible if the three vectors

c˙, r˙, r {displaystyle mathbf {dot {c}} ;,;mathbf {dot {r}} ;,;mathbf {r} }

lie in one level, i.e. H. are linear dependent. The linear dependence of three vectors can be determined using the determinant of these vectors:

• The tangential levels along the straight line
  x ( in 0, in ) = c ( in 0) + in r ( in 0) {displaystyle mathbf {x} (u_{0},v)=mathbf {c} (u_{0})+v;mathbf {r} (u_{0})}
  are the same if

det(c˙(u0),r˙(u0),r(u0))=0{Displaystyle det(Mathbf {Dot {C}} (U_{0});,;Mathbf {Dot {R}} (U_{0}} (U_{0});,;Mathbf {R} (U_{0});=; 0}

.

A producing one that applies to which this means torsal .

• A control area
  x ( in , in ) = c ( in ) + in r ( in ) {displaystyle quad mathbf {x} (u,v)=mathbf {c} (u)+v;mathbf {r} (u)}
  Can be handled into a level if the Gauss curvature disappears for all points. This is exactly the case if

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

In every point ^[5] d. that is when every producer is a torsal. A handlebarable area therefore also means Torso .

Properties of a handlebarable area: ^[6]

• The generating represent a crowd of asymptotot lines. They are also a crowd of curvature lines.
• A lockable area is either a (general) cylinder or a (general) cone or a tangent area (area consisting of the tangents of a room curve).

The determinant condition for handling areas gives you a way to numerically determine a connection stent between two given lead curves. The picture shows an example of an application: connection stent between two ellipses (one horizontal, the other vertical) and its handling. ^[7]

There is an insight into the use of handicrafable areas in the CAD area Interactive design of developable surfaces ^[8]

A historical Overview of handling areas gives Developable Surfaces: Their History and Application ^[9]

Definition [ Edit | Edit the source text ]

At a cylindrical The standard surface is all generating parallel, i.e. H. All direction vectors

r ( in ) {displaystyle mathbf {r} (u)}

are parallel and with it

r˙( in ) = 0 . {displaystyle {dot {mathbf {r} }}(u)=mathbf {0} .}

With two parallels, all points have the same distance to the other.

At a not Cylindrical control surface are neighboring producing slipsters and there is a point on the one hand straight that has minimal distance from the other. In this case is

r˙( in ) ≠ 0 . {Displaystyle {Dot {DOT {R} }(u)Neq Mathbf {0} .}

Such a point is called Central point . The entirety of the central points form a curve that Line or Throat line or Size . ^[ten] The latter name describes the strict line of a single-shaped rotation hyperboloid (see below) very clearly.

• In the central point of a producing person, the amount of the Gausskelkümmung assumes a maximum ^[11] .

A cylindrical area has no central points and therefore no strict line, or vivid: no waist. In the case of a (general) cone surface, the strict line/waist degenerates to a point, the cone tip.

Parameter presentation [ Edit | Edit the source text ]

In the following considerations it is assumed that the control area

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

is not cylindrical and sufficiently differentiated, more precisely:

r˙( in ) ≠ 0 {displaystyle {dot {mathbf {r} }}(u)neq mathbf {0} quad }

and for the sake of simplicity | r ( in ) | = first {displaystyle quad |mathbf {r} (u)|=1 }

is.

The last capacity has the advantage that

r ⋅ r˙= 0 {displaystyle quad mathbf {r} cdot {dot {mathbf {r} }}=0quad }

is what makes bills very simplified. In the case of concrete examples, this property is usually not fulfilled at first. But what can be corrected by standardization.

Two neighboring producing

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

x ( in 2) = c ( in + D in ) + in 2r ( in + D in ) {Displaystyle Mathbf {x} (V_{2})=Mathbf {C} (U+Delta U)+V_{2};Mathbf {R} (u+Delta u)

At the end of the considerations then goes

D in → 0 {displaystyle Delta uto 0}

. That is why the following linear approximations (you replace the curve in the vicinity with its tangent) are useful:

c ( in + D in ) ≈ c ( in ) + D in c˙( in ) {Displaystyle mathbf {C} (u+delta u) Approx mathbf {C} (u)+Delta u; {dot {mathbf {C}}} (u)}

r ( in + D in ) ≈ r ( in ) + D in r˙( in ) {Displaystyle mathbf {r} (u+delta u) Approx mathbf {r} (u)+Delta u; {dot {mathbf {r}}} (u)}

.

Spacer

The square of the distance between two points of the straight

l1( in 1) = c + in 1r {displaystyle mathbf {l} _{1}(v_{1})=mathbf {c} +v_{1};mathbf {r} }

l2( in 2) = c + D in c˙+ in 2( r + D in r˙) {Displaystyle Mathbf {L} _{2}(V_{2})=Mathbf {c} +Delta U;{Dot {Dot {Dot {Dot {Dot {C {C} }+V_{2};(Matbf {R} +Delta U; DOT {MATHBF {R} }))QUAD }

is

D ( in 1, in 2) = (( in 2– in 1) r + D in ( c˙+ in 2r˙) )2 . m Discussion ylesves of the empals—LPP. Ex. 2 Refane tuk . p.M .M Phil.

Parameter of the central point

The distance becomes minimal when the function

D ( in first , in 2 ) {displaystyle D(v_{1},v_{2})}

becomes minimal. And this is the case if the 1st partial derivations are zero:

D v1= – 2 (( in 2– in 1) r + D in ( c˙+ in 2r˙) )⋅ r {Displaystyle D_{V_{V_{1}}=-2{Big (} v_{2}-V_{1});Mathbf {R} +Delta U({Dot {Dot {Dot {Dot {C{C {C }}+V_{2}+V_{2}; ;{Dot {mathbf {R} }){Big )}Cdot Mathbf {R} }

 =−2(v2−v1+Δuc˙⋅r)=0 ,{Displaystyle =-2{Color {magenta}(V_{2}-V_{1}+Delta U;{Dot {Dot {Dot {Dot {CDF {C} }} CDOT MATHBF {R}=0 ,}

D v2= 2 (( in 2– in 1) r + D in ( c˙+ in 2r˙) )⋅ ( r + D in r˙) {Displaystyle D_{V_{V_{2}}=2{Big ( }(v_{2}-V_{1});Mathbf {R} +Delta U({Dot {Dot {Dot {Dot {MATBF {C {C }}+V_{2}; {Dot {Mathbf {R} }){Big )}Cdot (Mathbf {R} +Delta U;{Dot {Dot {Dot {R} })

 =2(v2−v1+Δuc˙⋅r+Δu2(c˙⋅r˙+v2r˙2))=0 .{Displaystyle =2{Big (}{Color {magenta}V_{2}-V_{1}+Delta U;{Dot {Dot {Dot {Dot {C {C} }}Cdot Mathbf {R} }+{Delta U}^{2} ({dot {mathbf {c} }}cdot {dot {r} }+V_{2}{dot {dot {dot {r} } } } } } } } }{big )}=0 .}=0

From this system of equations for

in first , in 2 {displaystyle v_{1},v_{2}}

follows for

D in → 0 {displaystyle Delta uto 0 }

:

• 
  in 1= in 2= – c˙⋅r˙r˙2 . {Displaystyle quad V_{1}=V_{2}=-{Frac {{Dot {Dot {Dot {CDF {C} } CDOT {DOT {DOT {RT } }{{Dot {Dot {Dot {Dot {R} }}^{2 }}}} .}
  

Parameter presentation

The parameter representation of the strict line is therefore

• x(u)=c(u)−c˙(u)⋅r˙(u)r˙2(u)r(u) .{Displaystyle Mathbf {x} (u)=Mathbf {c} (u)-{Dot {Dot {Dot {DOT {DOT {DOT {C) }(u)cdot {Dot {Dot {R) }(u)}{{Dot {Mathbf {R} }}^{2}(u)}}; Mathbf {R} (U)

  

Double control surfaces

Located on the intelligent hyperboloid and the hyperbolic paraboloid two Trops of straight lines. A strict line belongs to every crowd. The two strict lines collapse when the rotation hyperbolod is single.

Examples [ Edit | Edit the source text ]

1) Single-up rotation hyperboloid

x ( in , in ) = (cos⁡usin⁡u0)+ in ⋅ (−sin⁡ucos⁡uk) , {displaystyle mathbf {x} (u,v)={begin{pmatrix}cos u\sin u\0end{pmatrix}}+vcdot {begin{pmatrix}-sin u\cos u\kend{pmatrix}} , }

The central points all have the parameter

in = 0 {displaystyle v=0}

, d. H. The strict line is the unit circuit in the X-Y level.

2) Rades Konoid

With a straight conoid, the axis is the common plain of all the producers.
(In general: a pair of points of two wind slate straight is the shortest distance if its connection is the common plenty of the straight line.)

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

Examples of straight konoids are the hyperbolic paraboloid

With = x and {Displaystyle z = xy}

And the spiral area.

3) torso

Each from the general cylinder and cone different handlebarable control surface (torse) is a tangent area, i.e. H. The entirety of the producers of the control surface consists of the crowd of tangents of a given curve

c {displaystyle gamma }

. (In the picture, the curve is a screw line. This creates a screw oral.) Generally the following applies

The strict line one through a curve c {displaystyle gamma }

the tangent area generated is the curve c {displaystyle gamma }

self ^[twelfth] .

4) Möbiusband

For the description of a Möbiusband specified above is

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

,

r ( in ) = ( cos ⁡ in cos ⁡ 2 in , cos ⁡ in sin ⁡ 2 in , sin ⁡ in ) T . {Displaystyle Mathbf {r} (u) = (cos ucos 2u, cos usin 2u, without u)^{t}.}

(To the picture: So that the strict line is completely on the area shown, the tape was widened.)
The direction vector

r {displaystyle mathbf {r} }

In this case it is already a unit vector, which significantly simplifies the invoice.

The parameter of the respective central point results

in = 4cos⁡u1+4cos2⁡u{displaystyle v={frac {4cos u}{1+4cos ^{2}u}}}

and finally the parameter representation of the strict line

x ( in ) = 11+4cos2⁡u(cos ⁡ ( 2 in ) , sin ⁡ ( 2 in ) , – 2 sin ⁡ ( 2 in ) ) . {displaystyle mathbf {x} (u)={frac {1}{1+4cos ^{2}u}};{big (}cos(2u),sin(2u),-2sin(2u){big )} .}

It is easy to see that this curve in the level

2 and + With = 0 {displaystyle 2y+z=0}

lies.
To show that this level even curve

an ellipse with a center ( – 25, 0 , 0 ) {displaystyle (-{tfrac {2}{5}},0,0)}

and the half axles a = first , b = 35{displaystyle a=1,b={tfrac {3}{5}}}

is,

one shows that the X and Y coordinates are the equation

(x+25)2(35)2+ y215= first {displaystyle {tfrac {(x+{tfrac {2}{5}})^{2}}{({tfrac {3}{5}})^{2}}}+{tfrac {y^{2}}{tfrac {1}{5}}}=1 }

fulfill. So the floor plan of the strict line is an ellipse and thus the strict line as a parallel projection.

The strict line is easier through the parameter display

x ( t ) = f0+ f1cos ⁡ t + f2sin ⁡ t {displaystyle mathbf {x} (t)=mathbf {f} _{0}+mathbf {f} _{1}cos t+mathbf {f} _{2}sin t }

with

f0= ( – 25, 0 , 0 ) T, f1= ( 35, 0 , 0 ) T, f2= ( 0 , – 15, 25) T{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).

You can two handlebarable control surfaces along one straight

g {displaystyle g}

or.

h {displaystyle h}

cut off and put them together so that

g {displaystyle g}

and

h {displaystyle h}

A common just the composite area with a new common tangential level of this.

In the case of a non -handling and a handlebarable control area, the area composed along the common generators cannot be differentiated. The common generator is visible as an edge, with the edge of different points of the producing significantly clear. In the case of two non -lockable control surfaces, the area composed can be differentiated along the common generators, but it is generally not.

Regular areas can be used not only in mathematics, but also outside of it in constructions and engineering work. A good example of this is the work of the architect/mathematician Antoni Gaudí. The vault of the La Sagrada Família describes several hyperboloids, hyperbolic paraboloids and helicoids. ^{[13] [14]}

• Manfredo P. do Carmo: Differential geometry of curves and areas . Springs-Publising, 2013, 2013, ISBN 978-3-322-85498-0, P. $ 132,147
• G. White: Curves and Surfaces for Computer Aided Geometric Design . Academic Press, 1990, ISBN 0-12-249051-7
• D. Hilbert, S. Cohn-Vossen: Vivid geometry . Springs-Publising, 2013, 2013, ISBN 978-3-6662-36685-1, p. 181
• W. Kühnel: Differentialgeometrie . View, 2003, isbn 3-528-17289-4
• H. Schmidbauer: Pick -up surfaces: a design apprenticeship for practitioners . Springs-Publising, 2013, 2013, isbn 978-3-642-47353-1

1. ↑ D. B. 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, S. 228.
2. ↑ Rule . In: Jacob Grimm, Wilhelm Grimm (ed.): German dictionary . Band 14 : R – slate – (VIII). S. Hirzel, Leipzig 1893 ( woerterbuchetz.de ).
3. ↑ G. White: Curves and Surfaces for Computer Aided Geometric Design , Academic Press, 1990, ISBN 0-12-249051-7, S. 250
4. ↑ W. Wunderlich: Via a handlebarable Möbiusband , Monthly books for Mathematics 66, 1962, pp. 276–289.
5. ↑ W. Kühnel: Differentialgeometrie , S. 58–60
6. ↑ G. White: S. 380
7. ↑ Cad-Skript. (PDF; 2,9 MB) S. 113
8. ↑ Tang, Bo, Wallner, Pottmann: Interactive design of developable surfaces (PDF; 3,3 MB) 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, two: 10.1007/s00004-011-0087-z
10. ↑ W. Kühnel: Differentialgeometrie , Views, 2003, ISBN 3-528-17289-4, P88.
11. ↑ M. P. do Carmo: Differential geometry of curves and areas , JOURTER-PBlegly, 2013, 2013, ISB, 3322828 $72, Plimes
12. ↑ W. Haack: Elementary differentialgeometries , Springr-Publising, 2013, 2013, ISBR 30348699509, S. 32
13. ↑ above Enjoy geheimnis . Southgerman newspaper
14. ↑ About regular areas in the “Sagrada Familia”. Scienceblogs