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In a court scene in a movie, an eyewitness reported that he had eye contact with "the whole bus" during an event. A lawyer challenged this statement, saying "you can only observe the side of the bus that is facing towards you, which can never exceed 50%". This got me thinking:

Is it true that one can never view more than half of the surface area of any convex 3D object at the same time (without using optical devices like mirrors)? My thought is yes, since if the object is complete (with no holes), if you have a tiny surface area facing one direction, there must be another surface facing the exact opposite direction. But I can't proof this idea beyond "intuitive".

Does the surface of the object also have to be continuous?

kevin
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    If you look at a tetrahedron from a point somwhere near an vertex, you can see $3$ out of $4$ faces. – achille hui Apr 23 '16 at 16:47
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    Intuitively, for a line of sight parallel to the unit vector $\vec{n}$, one might "observe" 50% of the flux $\int \vec{n} \cdot \mathrm{d}\vec{A}$. –  Apr 23 '16 at 17:08
  • If we want to be picky about it, normally one can only see the surface of a bus and those parts of the interior that are visible through windows. Some of the visible surface may include the outer surface of the windows on the "other side" of the bus. The most reasonable interpretation of seeing the whole bus is that there are no other objects (aside from the bus itself) obscuring your view of the bus, and the lawyer's challenge is nonsensical unless there is some particular relevant thing about the bus that cannot be observed from all directions. – David K Apr 23 '16 at 19:50
  • @DavidK that was the lawyer's point in the movie: somebody could have jumped off the bus on the other side and it would not be visible to the eyewitness. – kevin Apr 23 '16 at 19:58
  • How would they have jumped off the bus? Were they hanging onto something on the outside (below the level of the windows), and let go? If they jumped off from inside the bus, they had to go through a door or window. One can easily observe more than 50% of those combined apertures at any one instant. Even if they jumped off the outside, more than 50% of the area they could jump from would have rendered them visible (in some cases, through a window), so I still call the "50%" a completely specious number. – David K Apr 23 '16 at 20:16
  • On second thought, heavily tinted windows could make a difference. It depends on the bus, then. – David K Apr 23 '16 at 20:17

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I'm assuming that by how much you can see of the object, you mean the maximum surface area projected onto a plane. In that case, 2D should suffice because you could always rotate or add depth to make a prism.

Take an isosceles triangle, point down, projected down onto a line parallel to the opposite side (think of setting it on the "ground" and then flipping it exactly upside down). Fix the perimeter $P = 1$. Then let the length of the parallel side be $\ell$. The length of the sides of the triangle that are facing the line is then $1- \ell$ and the fraction facing the line is $\frac{1- \ell}{1}$. As you make $\ell$ smaller, you get closer and closer to $100%$% of the triangle being "seen" by the line.

Replace the triangle with a cone and it still works. So for a really "sharp" cone, you can see almost $100%$% of it (if you're exactly facing the point).

Michael Klein
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