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Definition of A Non-Linear Reflection?

I am a recently graduated highschool senior and have been reconsidering a question I had been investigating in the past. A question that none of my teachers seemed to think was possible. I was considering what a reflection over a non-linear base would look like, and its actually taken me awhile to "define" what I think would be considered reflection, since its reversable. Say I want to reflect the line $\displaystyle g(x) = (1/3)*x$ over the parabola $\displaystyle f (x) = x^2 + 2$ and called the resulting graph $\displaystyle R(x)$. The way I defined it is the following: At any point $\displaystyle x$ on $\displaystyle f (x)$ construct a line normal to $\displaystyle f (x)$, and call this line $\displaystyle N(x)$ . Then determine where $\displaystyle N(x)$ intersects $\displaystyle g(x)$ and call this point $\displaystyle P$ . Construct a line tangent to $\displaystyle f (x)$ at $\displaystyle x$ and call it $\displaystyle T(x)$. Finnally, reflect evey point $\displaystyle (P , g( P ) ) $ over every corresponding line $\displaystyle T(x)$ to arrive at the equation of $\displaystyle R(x)$. I used this definition of non-linear reflections and another process using a function for the intersection point to derive the equation of $\displaystyle R(x)$ in terms of $\displaystyle x$ alone. I have digitally generated all my work, and I was wondering if anybody is willing to take a look at my work and give me some feed back. I'm not expecting anybody to read it over word for word, just a quick over-view so I can get some feed back as to weather this could be consider a correct definition. I will attach a few preview pictures of my work to this thread, the first picture being a reflection of the x-axis over $\displaystyle f(x)$ as defined, and the last picture is a reflection of $\displaystyle g(x)$ over $\displaystyle f (x)$ as defined [assuming that I didnt make any simple algebra mistakes] . If you are willing to look at the whole series of my work, email me at mfetch22@yahoo.com and I'll be glad to send it to you. Thanks in advance.

NOTE: The attached images are not in the proper order and have pages missing between them. If one slide says "on the next page I....." dont expect it to be on the next page. You need to whole set of images for it to work like that, the forum file attachment limit is 5, so I put the most important slides on this thread