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Gaming Cobra
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    I've edited the tags; your question has nothing to do with any of tags you've had attached at first. – mrtaurho Feb 01 '20 at 22:07
  • Good question, but it depends on the rotations you make. In the example of the cylinder, you can make one rotation of 180$^o$ (swapping flat ends) and there will be no difference in the shadow. You can also make any rotation about the axis through the centre of the flat ends without any change in the shadow. You therefore need to be more specific about the kinds of rotations you will allow. – tomi Feb 01 '20 at 22:11
  • $B$ could be a ball with a tiny bump on top, and if you make small rotations the bump will have no effect on the shadow cast by $B$. – littleO Feb 01 '20 at 22:11
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    Consider the how the Steinmetz solid is constructed, you can have as many arbitrary orientations as you want. – Lee David Chung Lin Feb 01 '20 at 22:11
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    @littleO I think the OP has to restraint his quest to convex shapes. – Jean Marie Feb 01 '20 at 22:27
  • @JeanMarie I agree the question is more interesting if $B$ is restricted to be convex. – littleO Feb 01 '20 at 22:31
  • A cilinder works for all your cases. But what if $B$ is compact and convex and every rotation has a perfectly round shadow? Does $B$ then have to be a sphere? – Magdiragdag Feb 01 '20 at 22:52
  • @ Magdiragdag The answer is yes : see https://mathoverflow.net/q/39127 – Jean Marie Feb 01 '20 at 23:24
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    I have taken the liberty to change your former title into a more informative one (it is important to drain more people to look at your question). I wish you don't object... – Jean Marie Feb 01 '20 at 23:30
  • Thank you for your changes. I am confused is the answer yes or no? – Gaming Cobra Feb 02 '20 at 02:06
  • @JeanMarie Could you further explain why it is yes and try to talk with other people here and see which is the correct response? – Gaming Cobra Feb 02 '20 at 02:46
  • Sorry I was sleeping (8:00 AM Central European Time). The answer to the title is "no" if we take the good orientations as examplified by my counterexample. But, if we modify the title into "... in any three (mutually) orthogonal directions, the answer would be "yes". – Jean Marie Feb 02 '20 at 08:07
  • @JeanMarie What about the answers to "Must B be a perfectly round ball? What if you are able to rotate B twice, getting now two new orientations, and all three shadows that you see are perfectly round and of the same diameter. Must B be a ball?" – Gaming Cobra Feb 02 '20 at 18:37
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    @GamingCobra: Do not vandalize your own question (or anyone else's!) by deleting its contents. – Blue Feb 04 '20 at 22:06

1 Answers1

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Remove a little cap to a sphere : you will be able to find three orthogonal directions in which projections will be perfect disks.

Jean Marie
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  • here is a question from ten years ago, one answer is by Thurston – Will Jagy Feb 01 '20 at 23:13
  • @Will Jagy Hi, Will. I would appreciate a reference. – Jean Marie Feb 01 '20 at 23:14
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    there is a final answer there from about 2016 by Anton Petrunin, he gives some references in Russian and translation refs. I'm guessing Joseph O'Rourke might have some favorite references https://math.stackexchange.com/users/237/joseph-orourke as he gets computers to draw surfaces; his faculty page http://cs.smith.edu/~jorourke/ – Will Jagy Feb 01 '20 at 23:32
  • @Will Jagy Thanks for the references. On my side, I can indicate you a good book, recently published : "Handbook of Geometric constraint systems principles" (what a title) Meera Sitharam, Audrey St. John, Jessica Sidman, CRC Press, 2019. For example, it is the first book where I see mentionned and used the Grassman-Pluecker relationships, that I used for the first time in a recent question (https://math.stackexchange.com/q/3493048) ; by the way, I am almost certain that there are different ways to solve it. – Jean Marie Feb 01 '20 at 23:44
  • Thanks for the title. It appears the campus libraries here have this Handbook just as an electronic resource, cannot just check it out: http://oskicat.berkeley.edu/record=b24726678~S1 – Will Jagy Feb 01 '20 at 23:54
  • Is this the only answer (to remove the little cap) Or do you have any other ideas? – Gaming Cobra Feb 02 '20 at 02:04
  • Somebody has said no and linked this: https://mathoverflow.net/q/39127 what do you think – Gaming Cobra Feb 02 '20 at 02:06
  • @WillJagy So what is the correct answer? – Gaming Cobra Feb 02 '20 at 02:46