Project a box from world to image plane

Question

I have a calibrated camera with extrinsincs and intrinsics parameters such that I can project a visible point in the world back to the image.

But I need to be able to deal with points that are also outside of what the camera can see.

1. how can I know if a point is visible by the camera? That would solve the issue when dealing with only one point.
2. how can I project a possibly only partially visible 3d box (8 points) back to the image?

My knowledge is limited and I cannot use a 3rd party framework to this end, rather need to implement it in python/C.