# Region created by locus of reflected lines

This may be more a physics topic but I feel like it is a math stack exchange topic.

Say you had a circular reflective surface with rays coming in parallel to the surface. They rays reflect with angle of incidence equaling angle of reflection. I've modeled the situation below for a different number of lines:

The red lines are the ray once it has bounced off the surface. I would also like to ignore the rays once they pass the y axis, i.e o only care about the red line segment from where the light ray first its the surface to where it first hits the y axis after reflection.It seems that these segments are beginning to form some region or special shape, like a very thing crescent moon that gets wider towards the middle, comes to a point, and then is symmetrical in nature on the other side of the y axis.I am clueless as to what this shape could be. ANy help? If you need I can give you the equations for the red lines and other needed information.

The shape i'm beginning to see looks like this: (crude drawing)

For your specific example, we can parameterize the semicircle as $$(4\cos t,-4\sin t)$$, $$0\lt t\lt\pi$$. The normal at a point on this semicircle has direction $$(-\sin t,-\cos t)$$, and so the direction of the reflected ray is $$2\left({(0,-1)\cdot(-\sin t,-\cos t)}\right)(-\sin t,-\cos t)-(0,-1) = (-\sin{2t},-\cos{2t}).$$ The reflected ray therefore lies on the line $$x\cos{2t}-y\sin{2t}=4\cos t$$. The envelope of these lines can be found by differentiating with respect to $$t$$ and solving the resulting system of linear equations for $$x$$ and $$y$$: $$x = 4\cos^3 t \\ y = -2(\cos{2t}+2)\sin t.$$