Calculating the “Belvedere Hull” of a Simple Planar Polygon

Question

As an informal motivation the problem, imagine a tower with polygonal footprint, that is located in a beautiful landscape, the "Belvedere Hull" is then related to the directions, in which one would have a view unobstructed by the tower (it is assumed that the tower has roof, so it is not possible to stand on top of it).

Problem:
given a simple, planar polygon, calculate the boundary of the union of all half-lines that do not contain inner points of the polygon, i.e. the "Belvedere Hull".

This problem seems to be related to the visibility from a point inside the polygon, or to the art gallery problem, so I would like to know if this problem has already been considered or, how to tackle it.

Generalizations can easily be envisaged, by either going to higher-dimensional spaces and/or, by replacing the polygon by other, not necessarily connected point-sets.

Answer 1

Your concept is called a weakly externally visible polygon in the literature.

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The above figure is from the chapter, "Visibility in the Plane" by Asano, Ghosh, Shermer, in The Handbook of Computational Geometry. Likely Ghosh's book is the best source:

Ghosh, Subir Kumar. Visibility algorithms in the plane. (Vol.2). Cambridge: Cambridge University Press, 2007.

The concept is studied extensively in the context of 3D graphics, e.g.,

Wang, Caoan, and Binhai Zhu. "Three dimensional weak visibility: Complexity and applications." Computing and Combinatorics. Springer Berlin Heidelberg, 1995. 51-60.