Translation surface (differential geometry)

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

• For two space curves ${\displaystyle c_{1},c_{2}}$ with a common point ${\displaystyle P}$, the curve ${\displaystyle c_{1}}$ is shifted such that point ${\displaystyle P}$ is moving on ${\displaystyle c_{2}}$. By this procedure curve ${\displaystyle c_{1}}$ 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.

Simple examples:

1. Right circular cylinder: ${\displaystyle c_{1}}$ is a circle (or another cross section) and ${\displaystyle c_{2}}$ is a line.
2. The elliptic paraboloid ${\displaystyle \;z=x^{2}+y^{2}\;}$ can be generated by ${\displaystyle \ c_{1}:\;(x,0,x^{2})\ }$ and ${\displaystyle \ c_{2}:\;(0,y,y^{2})\ }$ (both curves are parabolas).
3. The hyperbolic paraboloid ${\displaystyle z=x^{2}-y^{2}}$ can be generated by ${\displaystyle c_{1}:(x,0,x^{2})}$ (parabola) and ${\displaystyle c_{2}:(0,y,-y^{2})}$ (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.