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So, reflecting a function across the line y=x gives the inverse of the function. And I know that there's something about reflection matrices when reflecting across other straight lines, but what about reflecting across a curved function?

I imagined taking a given point, which I'll call $A$, on the original function, finding the nearest point to that one on the mirror function, which I'll call $B$, and reflecting $A$ across the line tangent to the mirror function at $B$. Then repeat this for every point in the line until you have your new function.

And I also know that this is a transformation, which means it can be defined algebraically. Now, is this something that has been talked about before?

Aiden Anderson
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  • What do you do when more than one point of the mirror function is equidistant to a point of your original? – RobinSparrow Sep 30 '24 at 21:59
  • The “circle of inversion,” is sort of an example of a reflection across a curve. Except, that there is a distance trasformation that goes with it. I don’t have the space to describe it fully here, but it is worth looking up. – user317176 Sep 30 '24 at 22:37
  • Welcome to Math.SE! ... "Now, is this something that has been talked about before?" Yes. A Math.SE site search should turn up quite a few related questions. ... Off the top of my head, I can think of this old answer of mine that discusses reflecting lines in a parabola. – Blue Sep 30 '24 at 23:36

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I played around with the same idea while I was in high school. My advice to you would be to come up with a practical example of such a transformation (draw a picture!) and see if you can generalize this into an algebraic form.

The answer to your question of whether or not this is water cooler gossip is no. It's not tantalizing enough! I can define a transformation of taking a point on a function in polar coordinates and scaling $r$ and $\theta$ by the values of the derivative and second derivative at a point on a different function. But why? Reflection across $y = x$ is practical and helps to solve problems. What does your transformation do?

You might look into inversive geometry! It solves many construction problems, including the one by Apollonius. https://en.wikipedia.org/wiki/Inversive_geometry

Alex
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