Certain continuous deformation of a trapezoid

Question

I have a problem related to the geometry. Let $ABCD$ be a trapezoid on the $R^2$ with, let's say, sides $AB$ and $CD$ being parallel, and let continuous functions $f:R^2\rightarrow R$ and $g:R^2\rightarrow R$ be given. Is it possible to find a formula for the continuous function $F:R^2\rightarrow R$ such that $F(AB)=f(AB)$, $F(CD)=g(CD)$ and $F(AD),F(BC)$ still being line segments? The existence of such a continuous function $F$ is intuitively rather certain but I really don't know how to obtain a formula for it. I would be grateful for any reply or hint.