A few weeks ago, I was taking the transformations unit in my precalculus class. I remember being intrigued by the reflections you could perform on a graph, and eventually derived this generalized linear axes reflection replacement. If you replace the x and y variables of a function with said replacement, it will reflect in the chosen linear axis.
However, after I derived that replacement, I wondered if it was possible to reflect in a curve rather than a line.
To create the linear axes replacement, I implemented some laws around flat mirrors such as how rays of light bounce off of them, along with two properties I observed around simpler reflections. These included the fact that the line of reflection is the 'invariant line,' and also the fact that the x and y coordinates of an original point O, and its reflected point R, always average out to lie on this line of reflection.
In physics, I remembered learning about circular and parabolic mirrors and lenses. So, I extrapolated those trends in tandem with the aforementioned observations to come up with a way to reflect in curves. I also found this post on the topic from a while ago, which talks about parabolas specifically.
I managed to implement the procedure with a semi circle without trouble, which can be found here. You may see that the point passing through the mid point of the curve of reflection is mapped to a second semi circle twice the radius of the original. I believe this is because in physics, you may have learned that the line of reflection for a circular mirror always points towards its center, and henceforth the normal lines intersecting at the one point result in a projection outwards in all directions.
This is where my first questions comes in. It is clearly possible for Desmos to map one point to an infinite number of points, however, it can not do the vice versa. As in, it seems that the points lying on the parabola in my example, which also lie on the larger semi circle, are not mapped to the midpoint of the curve of reflection. Is this a Desmos limitation, a larger logical flaw in my reasoning, or an incorrect question to ask in the first place because of some weird philosophical property of infinity?
In case you want to scrutinize the procedure I came up with, it can be found here.
My second questions relates to the implementation of this procedure on a parabola, here. I got to step 8, and tried to solve for n using the rational zero theorem, though did not make much progress. If it helps, Wolfram Alpha produced this result.