Translation surface (differential geometry)

In differential geometry a translation surface is a surface that is generated by translations: 
If both curves are contained in a common plane, the translation surface is planar (part of a plane). This case is generally ignored.
Simple "examples":
Translation surfaces are popular in descriptive geometry and architecture, because they can be modelled easily. <br>
In differential geometry minimal surfaces are represented by translation surfaces or as "midchord surfaces" (s. below).
The translation surfaces as defined here should not be confused with the translation surfaces in complex geometry.
Parametric representation.
For two space curves formula_15 and formula_16 with formula_17 the translation surface formula_18 can be represented by:
and contains the origin. Obviously this definition is symmetric regarding the curves formula_3 and formula_5. Therefore both curves are called generatrices (one: generatrix). Any point formula_22 of the surface is contained in a shifted copy of formula_3 and formula_5 resp.. The tangent plane at formula_22 is generated by the tangentvectors of the generatrices at this point, if these vectors are linearly independent.
If the precondition formula_17 is not fulfilled, the surface defined by (TS) may not contain the origin and the curves formula_27. But in any case the surface contains shifted copies of any of the curves formula_27 as parametric curves formula_29 and formula_30 respectively.
The two curves formula_27 can be used to generate the so called corresponding midchord surface. Its parametric representaion is
Helicoid as translation surface and midchord surface.
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 
has a "turn around shift" (German: Ganghöhe) formula_34. 
Introducing new parameters formula_35 such that
and formula_37 a positive real number, one gets a new parametric representation 
which is the parametric representation of a translation surface with the two "identical" (!) generatrices
The common point used for the diagram is formula_42.
The (identical) generatrices are helices with the turn around shift formula_43 which lie on the cylinder with the equation formula_44. Any parametric curve is a shifted copy of the generatrix formula_3 (in diagram: purple) and is contained in the right circular cylinder with radius formula_37, which contains the "z"-axis.
The new parametric representation represents only such points of the helicoid that are within the cylinder with the equation formula_47.
From the new parametric representation one recognizes, that the helicoid is a midchord surface, too:
where 
are two identical generatrices.
In diagram: formula_51 lies on the helix formula_52 and formula_53 on the (identical) helix formula_54. The midpoint of the chord is formula_55.
Advantages of a translation surface.
A surface (for example a roof) can be manufactured using a jig for curve
formula_5 and several identical jigs of curve formula_3. 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.
Establishing a parallel projection of a translation surface one 1) has to produce projections of the two generatrices, 2) make a jig of curve formula_3 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.
For a translation surface with parametric representation
formula_59
the partial derivatives of formula_60 are simple derivatives of the curves. Hence the mixed derivatives are always formula_61 and the coefficient formula_62 of the second fundamental form is formula_61, too. This is an essential facilitation for showing that (for example) a helicoid is a minimal surface.