The translation surfaces parametrized r(x,y)=(x,y,f(x)+g(y)) are obtained by translating two curves lying in orthonogal planes. What about the affine translation surfaces parametrized R(x,y)=(x,y,f(x)+g(y+ax)) for nonzero constant a ? Could the affine translation surfaces be geometrically described such as that of translation surfaces?
Why not f(ax+y)+g(y) ?
If you change your system of coordinates from x,y to X,Y (for instance X=ax+by and Y=cx+dy - a matrix representation with a rank 2 would be better), you have more generally f(X)+g(Y). If you fix b=0, as you did, then you fix one of the directions of translations.
I agree with the suggestion of Yan. Furthermore if one use the existing theory of geometric transformations, plus the conditions you would like to impose, can be easily have an answer.
Affine Transformations
http://people.cs.clemson.edu/~dhouse/courses/401/notes/affines-matrices.pdf
http://www.di.ubi.pt/~agomes/cg/teoricas/03e-transformations.pdf
http://www.cs.utexas.edu/~fussell/courses/cs384g-fall2010/lectures/lecture07-Affine.pdf
http://en.wikipedia.org/wiki/Affine_transformation
http://web.cs.wpi.edu/~emmanuel/courses/cs543/slides/lecture04_p1.pdf
http://www.mathworks.com/help/images/ref/affine3d-class.html
The replacement {x = x', y = y' + ax'} causes the surface to deform. This change of coordinates represents a Galilean motion. This replacement preserves the total curvature of the surface, even the Euclidean total curvature.
Good luck with your research.
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