In differential geometry a translation surface is a surface that is generated by translations:
For two space curves c 1 , c 2 {\displaystyle c_{1},c_{2}} with a common point P {\displaystyle P} , the curve c 1 {\displaystyle c_{1}} is shifted such that point P {\displaystyle P} is moving on c 2 {\displaystyle c_{2}} . Through this procedure, curve c 1 {\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:
Right circular cylinder: c 1 {\displaystyle c_{1}} is a circle (or another cross section) and c 2 {\displaystyle c_{2}} is a line. The elliptic paraboloid z = x 2 + y 2 {\displaystyle \;z=x^{2}+y^{2}\;} can be generated by c 1 : ( x , 0 , x 2 ) {\displaystyle \ c_{1}:\;(x,0,x^{2})\ } and c 2 : ( 0 , y , y 2 ) {\displaystyle \ c_{2}:\;(0,y,y^{2})\ } (both curves are parabolas). The hyperbolic paraboloid z = x 2 − y 2 {\displaystyle z=x^{2}-y^{2}} can be generated by c 1 : ( x , 0 , x 2 ) {\displaystyle c_{1}:(x,0,x^{2})} (parabola) and c 2 : ( 0 , y , − y 2 ) {\displaystyle c_{2}:(0,y,-y^{2})} (downward-open parabola). Translation surfaces are popular in descriptive geometry and architecture, because they can be modeled easily. 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 c 1 : x → = γ 1 ( u ) {\displaystyle \ c_{1}:\;{\vec {x}}=\gamma _{1}(u)\ } and c 2 : x → = γ 2 ( v ) {\displaystyle \ c_{2}:\;{\vec {x}}=\gamma _{2}(v)\ } with γ 1 ( 0 ) = γ 2 ( 0 ) = 0 → {\displaystyle \gamma _{1}(0)=\gamma _{2}(0)={\vec {0}}} the translation surface Φ {\displaystyle \Phi } can be represented by:
(TS) x → = γ 1 ( u ) + γ 2 ( v ) {\displaystyle \quad {\vec {x}}=\gamma _{1}(u)+\gamma _{2}(v)\;}
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