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Consider a 3D object, like a sphere or a cube. What's the minimum number of cameras required to fully capture all sides of that object? And at what angles should those cameras be placed?

My initial thoughts were that you might 4 placed at the vertex of a tetrahedron.

laggingreflex
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  • Just based on visualizing the problem and without offering a proof, I think that for an object with no concave surfaces that four cameras at a finite distance should be able to cover the object. If cameras at infinite distances are allowed, then two cameras on opposite sides of the object should be able to cover an object having no concave surfaces. – Barney Cowell Feb 13 '18 at 20:48
  • I ruled 2 camera system out as it might miss out the plane that's parallel to both cameras. Does having them at infinite distance overcome that restriction? – laggingreflex Feb 13 '18 at 20:56
  • @SamuelWeir It would appear that cameras at an infinite distance might prove quite impractical in a real scenario. – Jannik Pitt Feb 13 '18 at 20:58
  • I'm assuming "fully capture" essentially requires the normal of each surface has to have some component facing towards a camera, correct? As in, it doesn't count if your vision is parallel to the surface. – JMac Feb 13 '18 at 21:30
  • @SamuelWeir If "cover the object" requires all surfaces to have visual coverage normal to them, you would need 6 at finite distance AFAIK. – JMac Feb 13 '18 at 21:47
  • @Jmac - Under the conditions that I mentioned (no concave surfaces), I think that 4 cameras at a finite distance should cover it. If you can provide an example of an object with non-concave surfaces which requires 6 cameras, please do so. – Barney Cowell Feb 14 '18 at 00:15
  • @JannikPitt - Yes, I just mentioned the case of infinite cameras as an observation about the problem from a purely abstract mathematical standpoint: For a certain class of objects, two cameras can cover virtually all of the object, and in the limit that the cameras move farther and farther away from the object, their coverage of the object can approach 100%. – Barney Cowell Feb 14 '18 at 00:20
  • @laggingreflex - We're assuming that the cameras can be freely moved with respect to the object, right? So if there is a plane that is parallel to both cameras when the cameras are in certain position, then one can simply move the cameras (or rotate the object) so that plane is no longer parallel to both cameras. – Barney Cowell Feb 14 '18 at 00:23
  • @SamuelWeir If you had a sphere for example, and you considered "full capture" to include getting some reflection off of every surface, you couldn't do it with 4 cameras. Take this really bad picture I made for two cameras for example, and see the red line I drew for where your cameras would capture no information about the surface (although theoretically just an infinitely thin circle all around the sphere). Even with two more cameras, they would have another circle of missed space, and the two circles would intersect to make 2 missed points. – JMac Feb 14 '18 at 00:37
  • @JMac - I think that what you're saying is true IF the first two cameras are fixed on exactly the opposite sides of the object (or sphere in this case). In that case you would need four additional cameras (or actually three) in order to cover the circular ring area missed by the first two cameras. But if you put four cameras at the corners of a tetrahedron centered on the sphere, you can see that four cameras at those positions can cover the sphere. – Barney Cowell Feb 14 '18 at 00:54
  • @SamuelWeir 4 cameras works for almost every conceivable finite-sided object as far as I can think of, it would basically have to be infinitely complex to not work for some orientation of the 4. I believe that 4 cameras would still leave 2 poles though on a sphere. There are 2 places where a cameras wouldn't get an image from it, but they would see that it ends there. – JMac Feb 14 '18 at 01:12
  • @SamuelWeir, they don't even have to be at infinite distances. We use telecentric lenses in measuring machines to image one entire side of an object with a single camera –  Feb 14 '18 at 03:23
  • 1 camera if you are allowed to move it. Or 0 cameras if you draw it by hand. – Emil Feb 14 '18 at 07:09
  • 1 camera and some mirrors might do the trick too? – Emil Feb 14 '18 at 07:17
  • I am voting to close because this is a problem about geometry, not physics. – sammy gerbil Feb 14 '18 at 20:33

1 Answers1

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A good arrangement of cameras to photograph your sphere would be at the corners of a tetrahedron -- so four cameras. Or, six cameras could be arranged on the faces of a cube (much larger than the sphere, of course), or eight on the corners of a cube. Two cameras at infinity on opposite sides of the sphere would work too.

To photograph your cube, it would be possible to capture all 3D information by positioning two cameras on opposite ends of a line that passes through the center of the cube and opposite corners of the cube. The two cameras would not even need to be positioned at infinity to have a clear view of the cube's faces.

It's even possible to capture all 3D Information using only one camera: simply place mirrors at a few strategic points around the object.

As @SamuelWeir implied, concave portions complicate the problem. The shape of the object enters into the equation. In other words, there is not a unique answer to your question.

S. McGrew
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