I want to connect the cylinder to the half-sphere without changing their positions (they are separate objects with space between them). In reality the half-sphere and cylinder have a lot more faces than this simplified version. What is the most elegant way to do this? The connecting surfaces can be anywhere, as long as they are inside the hemi-sphere. In the end I want to 3D-print the model so that the cylinder is floating in the hemi-sphere exactly as it is positioned now.

One idea I came up with is to have a vertical plane be the connector, but the plane must not go through the cylinder, only touch it, and the plane must not go outside of the hemisphere, only touch it. But I don't know how to do that.

Here is my idea with the plane. If someone can tell me how to remove the section of the plane that is inside the cylinder and outside the hemisphere, then I am done: Update: Adding the difference Boolean modifier to the plane with the cylinder succeeds in removing the plane portion inside the cylinder, but I'm having difficulty removing the plane section outside the hemisphere. I'm also not sure if the plane is actually touching the cylinder or not during the 3D-print (if it is not, then the cylinder will fall). The new blend file is attached (second one).