fantastic problem, distorting mirror
Would this be considered a pluasible definition of a non-linear reflection? :
Say I have the line and I want to reflect it over the parabola , and call the resulting graph . At an arbitrary point on I construct a line normal to and call it . I determine the point on on at which intersects . Then I construct a line tangent to at and call it . Finnally, I reflect every point over every coresponding line to arrive at . 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 and and graphed the reflection defined above. I'll attach an image of it to this post.
Did you generate these reflections? They are absolutely fantastic! I considered looking at reflecting equations other then lines but I never took the time to generate the reflections! I love the trig function ones. Does the transformation matrix you posted above, when multiplied by the vector of and return these reflections for curved equations of also? Its so much simpler then my lengthy method. I knew transformation matricies could be used to rotate lines about the origin but I didnt know it could be applied to an explicit function. Great images in your attachments.