Dear mathoverflow community,

working on a visualization project I encountered a geometric problem, which I have not yet heard about and am interested in solving algorithmically. However a mere hint of what the problem might be called and what literature might deal with it would be very helpful as well:

In geometry a polygon P in the plane may be called monotone with respect to a line L, if every line orthogonal to L intersects P at most twice. I am however interested in the question, whether a polygon P is "monotone with respect to a point p", which I chose to mean that every straight line containing p intersects P at most twice.

To test this property I created a line from p through every vertex of P and counted the intersections with P. If none of those lines had more than two intersection points with P, I accepted P to be "monotone to p".

This property is easily extendible to 3D polyhedra: Every line through p shall intersect the polyhedron at most twice.

There are **three main questions** I am going to ask you about all this:

Does a name for this problem/property already exist? Which literature may deal with the property and recognition of polygon/polyhedra with this property.

In 3D, is it still sufficient to test every vertex to recognize a polyhedron with this property? (creating a line through p and every vertex each and counting the intersection points for the line with the polyhedron)

**Answered**2.1 If the polyhedron was limited to contain only axis aligned edges, would the per-vertex recognition suffice?

If an alternative algorithm comes to mind (especially if the answer to 2 turns out to be no) I am very glad to gather some ideas for algorithms for recognition of polyhedra with this property.

* Possible Algorithms *:

- Extend the polygon/polyhedron to contain p. Check if p is inside kernel.
- Apply projective transformation to polygon/polyhedron reducing problem to monotony with respect to a line.

Thanks in advance.

Kind regards,

Kilian Werner

point-monotoneorexternally star-shaped. $\endgroup$ – Joseph O'Rourke Sep 21 '16 at 14:15