projecting onto a mirror and back? Here's an illustration of what needs to be done:



A texture has to be projected from a point in space to a mirror (1 in the image) which will bounce the projection to a surface (2).

Here is the texture that is going to be projected although a real video file is going to be used.



The tricky thing is the mirror (the surface marked as 1 in the illustration) is not flat but curved as illustrated below:



The point is to see how the projected image or video will appear on the final surface (2) given a projection angle (fov) and a curved bounce mirror.
This is for a video for a project where I need to illustrate how a curved mirror deforms a projection from a video projector, laser projector and such and in the animation the shape of the mirror gradually changes (via shapekeys) which in turn changes the size and deformation of the projected image/video.

If Blender can do this somewhat accurately I can use it for simulations as well which is a plus because of the superior render quality compared to specialized CAD programs.

Also if this is possible with Blender and we have an answer then it makes Blender a powerful tool for people working with video projectors.
As has been said, Cycles isn't capable of this.  Luxrender, however, is.  You can read about it on this forum and elsewhere.  However, I'll say here that it integrates into Blender (previewing is even possible with the experimental API enabled as demonstrated below), and it handles refractive and reflective caustics very well.  It is a bit slower but the versatility is worth it for much more accurate physics than Cycles.

I have reproduced your scene by creating a Projector Spot Lamp (using your image) and a Mirror material on the reflector.  The reflector mesh has Subsurf and has Smooth Shading enabled.  The "screen" has the default Matte material:



Here I am morphing the plane from flat, to convex, to concave:



You must install Luxrender before opening the .blend: