Translation surface (differential geometry)

In differential geometry a translation surface is a surface that is generated by translations:

For two space curves $c_1, c_2$ with a common point $P$ , the curve $c_1$ is shifted such that point $P$ is moving on $c_2$ . Through this procedure, curve $c_1$ 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

File:Parabol-sf-sin.svg

File:schraubfl-ksf-sf.svg

Simple examples:

Right circular cylinder: $c_1$ is a circle (or another cross section) and $c_2$ is a line.
The elliptic paraboloid $\; z=x^2+y^2\;$ can be generated by $\ c_1:\; (x,0,x^2)\$ and $\ c_2:\;(0,y,y^2)\$ (both curves are parabolas).
The hyperbolic paraboloid $z=x^2-y^2$ can be generated by $c_1: (x,0,x^2)$ (parabola) and $c_2:(0,y,-y^2)$ (downwards open parabola).

Translation surfaces are popular in descriptive geometryH. Brauner: Lehrbuch der Konstruktiven Geometrie, Springer-Verlag, 2013,{{ISBN|3709187788}}, 9783709187784, p. 236Fritz Hohenberg: Konstruktive Geometrie in der Technik, Springer-Verlag, 2013, {{ISBN|3709181488}}, 9783709181485, p. 208 and architecture,Hans Schober: Transparente Schalen: Form, Topologie, Tragwerk, John Wiley & Sons, 2015, {{ISBN|343360598X}}, 9783433605981, S. 74 because they can be modelled easily.

In differential geometry minimal surfaces are represented by translation surfaces or as midchord surfaces (s. below).Wilhelm Blaschke, Kurt Reidemeister: Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie II: Affine Differentialgeometrie, Springer-Verlag, 2013,{{ISBN|364247392X}}, 9783642473920, p. 94

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

Parametric representation

For two space curves $\ c_1: \; \vec x=\gamma_1(u)\$ and $\ c_2:\; \vec x=\gamma_2(v)\$ with $\gamma_1(0)=\gamma_2(0)=\vec 0$ the translation surface $\Phi$ can be represented by:Erwin Kruppa: Analytische und konstruktive Differentialgeometrie, Springer-Verlag, 2013, {{ISBN|3709178673}}, 9783709178676, p. 45

:(TS) $\quad \vec x=\gamma_1(u)+\gamma_2(v) \;$

and contains the origin. Obviously this definition is symmetric regarding the curves $c_1$ and $c_2$ . Therefore, both curves are called generatrices (one: generatrix). Any point $X$ of the surface is contained in a shifted copy of $c_1$ and $c_2$ resp.. The tangent plane at $X$ is generated by the tangentvectors of the generatrices at this point, if these vectors are linearly independent.

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

The two curves $c_1,c_2$ can be used to generate the so called corresponding midchord surface. Its parametric representation is

: (MCS) $\quad \vec x=\frac{1}{2}(\gamma_1(u)+\gamma_2(v)) \; .$

Helicoid as translation surface and midchord surface

File:Wendelfl-sf.svg

File:Wendelflaeche-sfl.svg

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

: $\vec x(u,v)= (u\cos v,u\sin v, kv)$

has a turn around shift (German: Ganghöhe) $2\pi k$ .

Introducing new parameters $\alpha, \varphi$ J.C.C. Nitsche: Vorlesungen über Minimalflächen, Springer-Verlag, 2013, {{ISBN|3642656196}}, 9783642656194, p. 59 such that

: $u=2a\cos\left(\frac{\alpha-\varphi} 2 \right)\ , \ \ v=\frac{\alpha+\varphi}{2}$

and $a$ a positive real number, one gets a new parametric representation

$\vec X(\alpha,\varphi)= \left (a\cos\alpha + a\cos \varphi \; ,\; a\sin\alpha + a\sin \varphi\; ,\; \frac{k\alpha}{2}+\frac{k\varphi}{2}\right )$

::: $=(a\cos\alpha , a\sin\alpha , \frac{k\alpha}{2} ) \ +\ (a\cos\varphi , a\sin\varphi ,\frac{k\varphi}{2} )\ ,$

which is the parametric representation of a translation surface with the two identical (!) generatrices

: $c_1: \; \gamma_1=\vec X(\alpha,0)=\left(a+a\cos\alpha , a\sin\alpha , \frac{k\alpha}{2} \right) \quad$ and

: $c_2: \; \gamma_2=\vec X(0,\varphi)=\left(a+a\cos\varphi , a\sin\varphi ,\frac{k\varphi}{2} \right)\ .$

The common point used for the diagram is $P=\vec X(0,0)=(2a,0,0)$ .

The (identical) generatrices are helices with the turn around shift $k\pi\;,$ which lie on the cylinder with the equation $(x-a)^2+y^2=a^2$ . Any parametric curve is a shifted copy of the generatrix $c_1$ (in diagram: purple) and is contained in the right circular cylinder with radius $a$ , which contains the z-axis.

The new parametric representation represents only such points of the helicoid that are within the cylinder with the equation $x^2+y^2=4a^2$ .

File:Wendelfl-sehnenmfl.svg

From the new parametric representation one recognizes, that the helicoid is a midchord surface, too:

: $\begin{align}
\vec X(\alpha,\varphi) & = \left(a\cos\alpha , a\sin\alpha , \frac{k\alpha}{2} \right) \ +\ \left(a\cos\varphi , a\sin\varphi ,\frac{k\varphi}{2} \right) \\[5pt]
& =\frac{1}{2}(\delta_1(\alpha) +\delta_2(\varphi))\ ,\quad
\end{align}$

where

: $d_1: \ \vec x=\delta_1(\alpha)=(2a\cos\alpha , 2a\sin\alpha , k\alpha ) \ ,\quad$ and

: $d_2: \ \vec x=\delta_2(\varphi)=(2a\cos\varphi , 2a\sin\varphi , k\varphi ) \ ,\quad$

are two identical generatrices.

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

Advantages of a translation surface

; Architecture:

A surface (for example a roof) can be manufactured using a jig for curve

$c_2$ and several identical jigs of curve $c_1$ . 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 $c_1$ 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 projections.

; Differential geometry:

For a translation surface with parametric representation

$\vec x(u,v)=\gamma_1(u)+\gamma_2(v) \;$

the partial derivatives of $\vec x(u,v)$ are simple derivatives of the curves. Hence the mixed derivatives are always $0$ and the coefficient $M$ of the second fundamental form is $0$ , too. This is an essential facilitation for showing that (for example) a helicoid is a minimal surface.

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. 259
W. Haack: Elementare Differentialgeometrie, Springer-Verlag, 2013, {{ISBN|3034869509}}, p. 140
C. Leopold: Geometrische Grundlagen der Architekturdarstellung. Kohlhammer Verlag, Stuttgart 2005, {{ISBN|3-17-018489-X}}, p. 122
D.J. Struik: Lectures on classical differential geometry, Dover, reprint ,1988, pp. 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