This question is related to Seeing something from only one angle means you have only seen (what?)% of its surface area at most?

Starting from a solid sphere, how can this object be (continuously) transformed so that a viewer (with a single eye) can see most of the surface?

There is no upper limit as we e.g. can transform the sphere to a cone pointing towards the viewer so that the only invisible part is the circular base of the cone and by increasing the height, the portion of this area goes to zero. (and the object moves away to infinity...)

But I somehow feel a bit unsatisfied as this is mathematical perfectly fine, but the angle between the surface normal and the direction towards the viewer goes to 90deg so even if there is nothing blocking the view, the viewer couldn't recognize anything on the surface.

So how does this answer change if we say that the viewing angle is limited?

An (almost) real use case would be a solar collector with a reflective surface so that light that comes in too flat will be rejected and we want to collect as much energy as possible.

Some Clarification: We have a sphere so without restricting the viewing angle, we see half of the surface. However, at C, the viewing angle beta is almost 90deg, so the resolution of our image of this region gets arbitrary small. So I'd like to select an arbitrary alpha (in the drawing ~60deg) after which a point is no longer considered visible. In this drawing the surface between B and B' (which should be symmetrical) so for the sphere (and this distance to the viewer), around 20% of the surface should be considered visible (or recognizable).

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closed as unclear what you're asking by John Rennie, honeste_vivere, Yashas, Jon Custer, ZeroTheHero Aug 16 '17 at 21:43

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  • $\begingroup$ -1. Not clear how your question differs from the one cited by you. For example, how is the viewing angle limited in the case of the solar collector? In the general case, the obvious answer is that if a part of the surface is not within the viewing angle then it cannot be seen. $\endgroup$ – sammy gerbil Aug 13 '17 at 9:33
  • $\begingroup$ How is it not clear? A point on the surface is considered visible if there is there is nothing blocking the view and the angle between the surface normal at this point an the line towards the viewer is less than a threshold (e.g. 20deg). $\endgroup$ – FooBar Aug 13 '17 at 9:49
  • $\begingroup$ That issue is addressed in the other question. When this angle is close to 90 degrees all points on the surface are compressed into a very small angle. They are in theory still visible. It is only practical reasons which make them not invisible, such as surface roughness (one point hidden behind another) or the limited angular resolution of the eye (two points indistinguishable because the angle between them is too small). $\endgroup$ – sammy gerbil Aug 13 '17 at 10:04
  • $\begingroup$ And no I ask how the answer changes if we restrict the points to those which have a limited viewing angle. Why do you think that this not a viable question? Especially on physics.se (and not math.se) this sounds (at least to me) like an important restriction. $\endgroup$ – FooBar Aug 13 '17 at 10:18
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    $\begingroup$ An alternative criterion is the critical angle for total internal reflection. If the sphere is covered with a transparent layer of index $n$, the maximum value of $\alpha$ is the critical angle for TIR. $\endgroup$ – sammy gerbil Aug 13 '17 at 18:51