A "matched set" of synchronized anamorphic cameras can be used to create the illusion of the effect you describe.
Another approach would be to use a reflecting (mirror) lens with the required distortion to capture the panorama and also be used to undo the distortion to project the image correctly to a viewer. By looking (or recording) directly along the axis at the apex of the cone, a panorama is revealed.
She's wearing a little black dress and drinking merlot from a shatterproof wineglass one of her viewers sent her after she'd broken a real one on camera.
She makes almost $400 in the 45 minutes I'm with her, and she doesn't do much besides talk to me (offscreen) about camming.
Ideally such a system should ideally be able to capture at least 200 lines per second, and should cost less than $500.
Film photography (such as the "Spinner 360" camera) is not a good option.
This is why I'm so particular about using a Strip Camera similar to "photo finish" cameras used for racing finish lines.
While you wish to capture the 360° panorama, you don't say what you want to do with the captured image.
The end of the captured record will not be seamless with the beginning of the sweep.
There will be a mis-match unless the entire panorama can be captured simultaneously.
Most of her viewers right now are her "friends," who seem happy just to hang out, listen to her talk, and reminisce about their shared stories. "People still come into my room asking about the coffee stand," says Eevie.
"Everyone misses it." Eevie got her start camming by setting up her laptop inside the bikini barista drive-through espresso stand she was working at, which is apparently a novelty to people around the world. Like, 'Holy shit, there's a girl in her underwear in public.'" The rules have changed since then.
My ultimate goal is to set up some sort of digital strip camera rig for taking stereoscopic 360° panoramas.