View source for Translation surface (differential geometry)

{{Short description|Surface generated by translations}}
[[File:Parabol-sf-def.svg|thumb|Translation surface: definition]]
In [[differential geometry]] a '''translation surface''' is a [[Surface (mathematics)|surface]] that is generated by [[Translation (geometry)|translations:]] 
* For two [[space curve]]s <math>c_1, c_2</math> with a common point <math>P</math>, the curve <math>c_1</math> is shifted such that point <math>P</math> is moving on <math>c_2</math>. By this procedure curve <math>c_1</math> generates a surface: the ''translation surface''.

If both curves are contained in a common plane, the translation surface is planar (part of a plane). This case is generally ignored.

[[File:Parabol-ezh-sf.svg|500px|thumb|ellipt. paraboloid, parabol. cylinder, hyperbol. paraboloid as translation surface]]
[[File:Parabol-sf-sin.svg|thumb|translation surface: the generating curves are a sine arc and a parabola arc]]
[[File:schraubfl-ksf-sf.svg|thumb|Shifting a horizontal circle along a helix]]
Simple ''examples'':
#[[Right circular cylinder]]: <math>c_1</math> is a [[circle]] (or another cross section) and <math>c_2</math> is a line.
#The ''elliptic'' [[paraboloid]] <math>\; z=x^2+y^2\; </math> can be generated by  <math> \ c_1:\; (x,0,x^2)\ </math> and <math>\ c_2:\;(0,y,y^2)\ </math> (both curves are [[parabola]]s).
#The ''hyperbolic'' paraboloid <math>z=x^2-y^2</math> can be generated by <math>c_1: (x,0,x^2)</math> (parabola) and <math>c_2:(0,y,-y^2)</math> (downwards open parabola).

Translation surfaces are popular in [[descriptive geometry]]<ref>H. Brauner: ''Lehrbuch der Konstruktiven Geometrie'', Springer-Verlag, 2013,{{ISBN|3709187788}}, 9783709187784, p. 236</ref><ref>Fritz Hohenberg: ''Konstruktive Geometrie in der Technik'', Springer-Verlag, 2013, {{ISBN|3709181488}}, 9783709181485, p. 208</ref> and architecture,<ref>Hans Schober: ''Transparente Schalen: Form, Topologie, Tragwerk'', John Wiley & Sons, 2015, {{ISBN|343360598X}}, 9783433605981, S. 74</ref> because they can be modelled easily. <br>
In [[differential geometry]] minimal surfaces are represented by translation surfaces or as ''midchord surfaces'' (s. below).<ref>Wilhelm Blaschke, Kurt Reidemeister: ''Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie II: Affine Differentialgeometrie'', Springer-Verlag, 2013,{{ISBN|364247392X}}, 9783642473920, p. 94</ref>

The translation surfaces as defined here should not be confused with the [[translation surface]]s in [[complex geometry]].

== Parametric representation ==
For two space curves <math>\ c_1: \; \vec x=\gamma_1(u)\ </math> and <math> \ c_2:\; \vec x=\gamma_2(v)\ </math> with <math>\gamma_1(0)=\gamma_2(0)=\vec 0</math> the translation surface <math>\Phi</math> can be represented by:<ref>Erwin Kruppa: ''Analytische und konstruktive Differentialgeometrie'', Springer-Verlag, 2013, {{ISBN|3709178673}}, 9783709178676, p. 45</ref>
:'''(TS)''' <math> \quad \vec x=\gamma_1(u)+\gamma_2(v) \; </math>
and contains the origin. Obviously this definition is symmetric regarding the curves <math>c_1</math> and <math>c_2</math>. Therefore, both curves are called '''generatrices''' (one: [[generatrix]]). Any point <math>X</math> of the surface is contained in a shifted copy of <math>c_1</math> and <math>c_2</math> resp.. The [[tangent plane]] at <math>X</math> is generated by the tangentvectors of the generatrices at this point, if these vectors are [[linearly independent]].

If the precondition <math>\gamma_1(0)=\gamma_2(0)=\vec 0</math> is not fulfilled, the surface defined by '''(TS)''' may not contain the origin and the curves <math>c_1,c_2</math>. But in any case the surface contains shifted copies of any of the curves <math>c_1,c_2</math> as parametric curves <math>\vec x(u_0,v)</math> and <math>\vec x(u,v_0)</math> respectively.

The two curves <math>c_1,c_2</math> can be used to generate the so called corresponding '''midchord surface'''. Its parametric representation is
: '''(MCS)''' <math>\quad \vec x=\frac{1}{2}(\gamma_1(u)+\gamma_2(v)) \; .</math>

== Helicoid as translation surface and midchord surface ==
[[File:Wendelfl-sf.svg|thumb|Helicoid as translation surface with identical generatrices <math>c_1, c_2</math>]]
[[File:Wendelflaeche-sfl.svg|thumb|Helicoid as translation surface: any parametric curve is a shifted copy of the purple helix.]]
A [[helicoid]] is a special case of a [[generalized helicoid]] and a [[ruled surface]]. It is an example of a [[minimal surface]] and can be represented as a translation surface.
 
The helicoid with the parametric representation 
:<math>\vec x(u,v)= (u\cos v,u\sin v, kv)</math> 
has a ''turn around shift'' (German: Ganghöhe) <math>2\pi k</math>. 
Introducing new parameters <math>\alpha, \varphi</math><ref>J.C.C. Nitsche: ''Vorlesungen über Minimalflächen'',  Springer-Verlag, 2013,  {{ISBN|3642656196}}, 9783642656194, p. 59</ref> such that
:<math>u=2a\cos\left(\frac{\alpha-\varphi} 2 \right)\ , \ \  v=\frac{\alpha+\varphi}{2}</math>
and <math>a</math> a positive real number, one gets a new parametric representation 
* <math>\vec X(\alpha,\varphi)= \left (a\cos\alpha + a\cos \varphi \; ,\; a\sin\alpha + a\sin \varphi\; ,\; \frac{k\alpha}{2}+\frac{k\varphi}{2}\right )</math>
:::<math>=(a\cos\alpha  , a\sin\alpha  , \frac{k\alpha}{2} ) \ +\ (a\cos\varphi  , a\sin\varphi  ,\frac{k\varphi}{2} )\ ,</math>
which is the parametric representation of a translation surface with the two ''identical'' (!) generatrices
:<math>c_1: \; \gamma_1=\vec X(\alpha,0)=\left(a+a\cos\alpha  , a\sin\alpha  , \frac{k\alpha}{2} \right) \quad </math> and
:<math>c_2: \; \gamma_2=\vec X(0,\varphi)=\left(a+a\cos\varphi  , a\sin\varphi  ,\frac{k\varphi}{2} \right)\ .</math>
The common point used for the diagram is <math> P=\vec X(0,0)=(2a,0,0)</math>.
The (identical) generatrices are helices with the turn around shift <math>k\pi\;, </math> which lie on the cylinder with the equation <math>(x-a)^2+y^2=a^2</math>. Any parametric curve is a shifted copy of the generatrix <math>c_1</math> (in diagram: purple) and is contained in the right circular cylinder with radius <math>a</math>, which contains the ''z''-axis.
 
The new parametric representation represents only such points of the helicoid that are within the cylinder with the equation <math>x^2+y^2=4a^2</math>.

[[File:Wendelfl-sehnenmfl.svg|thumb|Helicoid as midchord surface of two identical generatrices (green helix).]]
From the new parametric representation one recognizes, that the helicoid is a midchord surface, too:

: <math>
\begin{align}
\vec X(\alpha,\varphi) & = \left(a\cos\alpha  , a\sin\alpha  , \frac{k\alpha}{2} \right) \ +\ \left(a\cos\varphi  , a\sin\varphi  ,\frac{k\varphi}{2} \right) \\[5pt]
& =\frac{1}{2}(\delta_1(\alpha) +\delta_2(\varphi))\ ,\quad
\end{align}
</math>
where 
:<math>d_1: \ \vec x=\delta_1(\alpha)=(2a\cos\alpha  , 2a\sin\alpha  , k\alpha ) \ ,\quad </math> and 
:<math>d_2: \ \vec x=\delta_2(\varphi)=(2a\cos\varphi  , 2a\sin\varphi  , k\varphi ) \ ,\quad </math> 
are two identical generatrices.

In diagram: <math>P_1: \delta_1(\alpha_0) </math> lies on the helix <math>d_1</math> and  <math>P_2: \delta_2(\varphi_0)</math> on the (identical) helix <math>d_2</math>. The midpoint of the chord is <math>\ M: \frac{1}{2}(\delta_1(\alpha_0) +\delta_2(\varphi_0))=\vec X(\alpha_0,\varphi_0)\ </math>.

== Advantages of a translation surface ==

; Architecture:
A surface (for example a roof) can be manufactured using a [[Jig (tool)|jig]] for curve
<math>c_2</math> and several identical jigs of curve <math>c_1</math>.  The jigs can be designed without any knowledge of mathematics. By positioning the jigs the rules of a translation surface have to be respected only.

; Descriptive geometry: 
Establishing a [[parallel projection]] of a translation surface one 1) has to produce projections of the two generatrices, 2) make a jig of curve <math>c_1</math>  and 3) draw with help of this jig copies of the curve respecting the rules of a translation surface. The contour of the surface is the envelope of the curves drawn with the jig.  This procedure works for orthogonal and oblique projections, but not for [[central projection]]s.

; Differential geometry:
For a translation surface with parametric representation
<math> \vec x(u,v)=\gamma_1(u)+\gamma_2(v) \; </math>
the [[partial derivative]]s of  <math> \vec x(u,v)</math> are simple derivatives of the curves. Hence the mixed derivatives are always <math>0</math>  and the coefficient <math>M</math> of the [[second fundamental form]] is <math>0</math>, too. This is an essential facilitation for showing that (for example) a helicoid is a minimal surface.

== References ==
<References />
* G. Darboux: ''Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal'', 1–4, Chelsea, reprint, 972, pp. Sects. 81–84, 218
* [[Georg Glaeser]]: ''Geometrie und ihre Anwendungen in Kunst, Natur und Technik'', Springer-Verlag, 2014, {{ISBN|364241852X}}, p.&nbsp;259
* W. Haack: ''Elementare Differentialgeometrie'', Springer-Verlag, 2013, {{ISBN|3034869509}}, p.&nbsp;140
* C. Leopold: ''Geometrische Grundlagen der Architekturdarstellung.'' [[Kohlhammer Verlag]], Stuttgart 2005, {{ISBN|3-17-018489-X}}, p.&nbsp;122
* D.J. Struik: ''Lectures on classical differential geometry'', Dover, reprint ,1988,  pp.&nbsp;103, 109, 184

== External links ==
* [https://www.encyclopediaofmath.org/index.php/Translation_surface Encyclopedia of Mathematics]

[[Category:Surfaces]]
[[Category:Differential geometry]]
[[Category:Differential geometry of surfaces]]
[[Category:Analytic geometry]]