One way is to use a [parabolic mirror](https://books.google.com/books?id=Vi-qCAAAQBAJ&pg=PA61&lpg=PA61&dq=parabolic+mirror+360). You photograph straight up into a curved mirror which gives you a distorted image with a 360° field of view. Software can then fairly simply undo the distortion. While the horizontal field of view is 360°, this approach can’t see straight up or down; the vertical field of view is only about 100°.
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