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).

Simple examples: Translation surfaces are popular in descriptive geometry[1][2] and architecture,[3] because they can be modelled easily.

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

can be represented by:[5] and contains the origin.

Obviously this definition is symmetric regarding the curves

Therefore, both curves are called generatrices (one: generatrix).

of the surface is contained in a shifted copy of

is generated by the tangentvectors of the generatrices at this point, if these vectors are linearly independent.

is not fulfilled, the surface defined by (TS) may not contain the origin and the curves

But in any case the surface contains shifted copies of any of the curves

can be used to generate the so called corresponding midchord surface.

Its parametric representation is A helicoid is a special case of a generalized helicoid and a ruled surface.

The helicoid with the parametric representation has a turn around shift (German: Ganghöhe)

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

The (identical) generatrices are helices with the turn around shift

Any parametric curve is a shifted copy of the generatrix

(in diagram: purple) and is contained in the right circular cylinder with radius

The new parametric representation represents only such points of the helicoid that are within the cylinder with the equation

From the new parametric representation one recognizes, that the helicoid is a midchord surface, too: where are two identical generatrices.

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

and several identical jigs of curve

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

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.

For a translation surface with parametric representation

are simple derivatives of the curves.

This is an essential facilitation for showing that (for example) a helicoid is a minimal surface.