EDIT (David White): This is really a comment to Gerhard's answer, not an answer in and of itself.
Thank you for your input Gerhard. A small clarification: what is desired is not two surfaces f3 $f_3$ and f4, $f_4$, but just one function that would undergo directional distortion and that would be some interpolation of f1 $f_1$ and f2 $f_2$ (to ensure tangentiality as f1 $f_1$ grows closer to f2) $f_2$).
Regarding the straw model. Are you using some discrete description (particles)? Where each point would undergo a different displacement u? $u$? If so, we can generate a surface f3 $f_3$ but we loose the closed-form formula for it, am I correct?