1. Parallax barrier display [link]: As used in the Nintendo 3DS, this is a display with a parallax barrier over the front of it, which blocks alternating columns of pixels from each eye's view. The result is that one eye gets to see the even columns, and other gets to see the odd columns.
Then, you can display separate images in the even columns and in the odd columns, each image being the image seen by the corresponding eye of a 3D scene, resulting in the illusion of a 3D image.
2a. Omnidirectional stereo speaker [link pending]: This a speaker that presents stereo audio to you no matter where you're standing in the space around it. It can actually provide stereo audio to an unlimited number of people standing and moving around it, with no control at all (active or passive), and using only two speaker drivers. It works by using a swirled radial array of horns for each speaker, with the two arrays being swirled in opposite directions and stacked on top of each other. The result is that, no matter where you're standing around this disc, the left-channel sound comes toward you from the left side of the disc, and the right-channel audio comes toward you from the right side. There's also a version with lots of individual speakers instead of horns from one speaker [link], which leads me to
2b. 360° stereoscopic 3D camera [link]: This is a radial array of cameras, all aimed outward, but alternately aimed left and right of the radial direction. They are aimed so far off the radial direction that the two cameras nearly on opposite sides of the circle both look in the same direction. The result is that you can take photos or video that can be viewed using a VR headset, and the viewer can look in any horizontal direction and get a stereoscopic 3D image (after the images are processed into a left circle and a right circle).
Description of new idea:
Take a parallax barrier display and wrap it into a cylinder.
Now, due to the angle of view changing more drastically across the visible width of the display compared to how little it does over the width of a flat parallax barrier display, it may be necessary to use more than two sets of pixel columns.
The only issue is that, as you go around the curve from the center of your visible area to the edge (or the limb, in astronomical terms) of your visible area of the display, the angle between your eyes and the line through the parallax barrier and the underlying pixels columns changes smoothly rather than discretely. This will cause some blurring between pixel columns, but this is probably acceptable. (The openings in the parallax barrier, if they are deep enough relative to their width, will naturally get narrower as the angle changes, maintaining the same visible width as the visible width of a pixel at that angle, but that 1-pixel width will not be aligned with the pixel columns in the underlying display, so horizontal blurring of two adjacent columns will result. It may be possible to design a specialized underlying display with specially aligned or spaced pixel columns to counteract
Anyway, when you stand in front of this cylindrical display, it works like any other parallax barrier display to provide a 3D-looking image by showing a different image to each of your eyes. But, as you walk around it, that effect turns out to continue over that greater (full 360°) range of azimuths. All the display has to do is provide, at every angle, an image of the 3D model it's displaying as if viewed from that angle.
Now, that sounds like it would require a lot of pixel column sets, if we're going to address, say, 360 separate views rather than the two views addressed in a traditional 2D parallax barrier display that's used by one user in one head position. But I think we can get away with addressing considerably fewer views, and using the binocular 3D effect to trick the brain into interpolating between those addressed views as the user moves around the display. This won't work if the two eyes connected to the same brain are within the same addressed view, so we need to decide on a maximum distance to view this display from, and then choose the number of addressable views based on that and the typical range of human eye separation. Viewers outside that maximum distance will not get any binocular 3D effect, and will only get jerky motion parallax as they walk around the display. But most volumetric display applications involve pretty close-range viewing, so I think this is okay.
Another limitation is that the display, as described so far, has no vertical parallax capability, so it has to be at the user's eye level. It might be possible to add a parallax barrier for the vertical axis, but I don't have time to think about that right now.