Cartesian parametrization: . When the generatrices are planes: . Cartesian equation (special case): . Partial differential equation: .

A translation surface is a surface that is the union of curves translated images of one another;
It is therefore the resulting surface of the translation of a curve (first generatrix) along another one (second generatrix); this definition is symmetric in the sense that the translation of the second generatrix along the first one results in the same surface.
 

For example, the surface  is obtained as the translation of the red curve  along the blue parabola  or the opposite.

A translation surface is therefore a special case of Darboux surface.
If we define the Minkowski sum of two subsets  and  of the space as the set of points M such that  where  describes  and  describes , then a translation surface can be defined as the Minkowski sum of two curves.

We get an equivalent definition by considering the surfaces geometric loci of the middles of the segment lines the ends of which describe two curves ("midsurface").

Examples:
    - the plane (case where the two generatrices are lines)
    - the cylinders (case where the first generatrix is a line)
    - the hyperbolic and elliptic paraboloids (the two generatrices are parabolas), only translation quadrics.
    - the right helicoid (the two generatrices are circular helices)
    - the Bohemian dome (the two generatrices are circles)
    - the egg box (the two generatrices are sinusoids)
    - the revolution of the sinusoid (the two generatrices are circular helices)
    - the Scherk surface
 
 

and the surfaces z = f(x) g(y)?
 

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© Robert FERRÉOL  2017