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.
Translation surface: definition

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

ellipt. paraboloid, parabol. cylinder, hyperbol. paraboloid as translation surface
translation surface: the generating curves are a sine arc and a parabola arc
Shifting a horizontal circle along a helix

Simple examples:

  1. Right circular cylinder: is a circle (or another cross section) and is a line.
  2. The elliptic paraboloid can be generated by and (both curves are parabolas).
  3. 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.