# Translation surface (differential geometry)

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

- For two space curves with a common point , the curve is shifted such that point is moving on . By this procedure curve 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.

Simple *examples*:

- Right circular cylinder: is a circle (or another cross section) and is a line.
- The
*elliptic*paraboloid can be generated by and (both curves are parabolas). - The
*hyperbolic*paraboloid can be generated by (parabola) and (downwards open parabola).

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

In differential geometry minimal surfaces are represented by translation surfaces or as *midchord surfaces* (s. below).[4]

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