From a viewpoint outside a circle in the plane, only part of the circle is visible, where a point on the circle is visible from the viewpoint if the line segment from the viewpoint to the point on the circle meets no other point of the circle. Call the set of points of the circle that are visible from a given viewpoint the partial view from the viewpoint of the circle. The partial views of enough viewpoints cover the circle (by compactness). What is the minimum number of viewpoints whose partial views cover the circle? Three. Jack up the dimension, and ask the same question about a sphere. The minimum number of viewpoints whose partial views cover the sphere is five. Why? Say there is a viewpoint over the North pole. Then the equator is not seen, no matter how far away is the the viewpoint. Likewise, from beneath the South pole a viewpoint cannot see the equator. So, from those two viewpoints the partial views see all of the sphere except for a (possibly very narrow) band around the equator. By the result for the circle, three more viewpoints are necessary to cover the band.
My question is, what is there in the mathematical literature that addresses the general question of determining the minimum number of viewpoints required to cover a given smooth shape in three-dimensional Euclidean space? For example, what is that minimum number for a torus? Six.