Would this be considered a pluasible definition of a non-linear reflection? :

Say I have the line $\displaystyle g(x)$ and I want to reflect it over the parabola $\displaystyle f (x)$, and call the resulting graph $\displaystyle R(x)$. At an arbitrary point $\displaystyle x$ on $\displaystyle f (x)$ I construct a line normal to $\displaystyle f (x)$ and call it $\displaystyle N(x)$. I determine the point on $\displaystyle P$ on $\displaystyle g(x)$ at which $\displaystyle N(x)$ intersects $\displaystyle g(x)$. Then I construct a line tangent to $\displaystyle f (x)$ at $\displaystyle x$ and call it $\displaystyle T(x)$. Finnally, I reflect every point $\displaystyle ( P, g(P) )$ over every coresponding line $\displaystyle T(x)$ to arrive at $\displaystyle R(x)$. This is just a thought I had, I dont think it has any applications, although the one thing it reminds me of is looking into the curved parabolic mirrors at the grocery store that help you see around corners, it takes a straight line frame of referance and "squeezes" is outward. Anyway I picked two random equations for $\displaystyle f (x)$ and $\displaystyle g(x)$ and graphed the reflection defined above. I'll attach an image of it to this post.