conical transform

Hi folks! would be awesome if I could get help!

I am trying to project a 3d animated model using a projector onto a Cone, so the cone would reflect the shape being projected on it to the sides. Now the cone is put in a cylinder, so the reflected light from the cone would be overlaid on the cylinder. I want the shape to be geometrically identical when projected on the cylinder to this original shape! There should be a kind of a transform matrix, or some kind of texture transform for such type of task right? I am using open gl, but I don’t know if that actually would make any difference…

THANKS A LOT if you can take a look at this!