I read once, in a book, about a lens that was an "inverse fisheye". Basically a hemispherical mirror mounted at one end of a glass tube.
I can't remember what it was called, nor can I find any record of its existance on the internet. Sadly, I no longer have the book to be able to reference it.
Has anyone heard of such a thing, and can name it, or better yet, got one, and some photos from it?
That's a one-shot 360° panoramic lens. There are a number of examples out there with different mirror shapes; two of the currently-active vendors are
They use different mirror shapes -- the 0-360 version is similar to your drawing, while the EGG is more like what would be left after a circle is swept out of a cylinder, so the effect (as a flat picture) would be different. The EGG is the only current one I could find that didn't have the mirror exposed and at the end of a long, delicate-looking stalk, but since I'm not looking for such a lens myself, I didn't spend a lot of time searching.
(When I wanted a 360° panorama in the film days, I rented a slit-shutter revolving camera. It opened the shutter and rotated the camera through a full circle while advancing the film -- sort of like a flatbed scanner for the world.)