# Monotone polygons (and polyhedra) with respect to a point

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:

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

2. 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)

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

3. 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 :

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

Kind regards,

Kilian Werner

• Perhaps reasonable terms are point-monotone or externally star-shaped. – Joseph O'Rourke Sep 21 '16 at 14:15
• Extending the polygon to contain $p$ and then checking to see if $p$ is in the kernel will not always work, because the portion removed as part of the extending could contain convolutions that cross a ray from $p$ several times. – Joseph O'Rourke Sep 22 '16 at 11:44
• @JosephO'Rourke Is this true? Even if the extension removes only faces that are entirely visible from p? – K. Werner Sep 22 '16 at 12:20
• I was assuming a particular meaning to "extending the polygon to contain $p$," I guess not your meaning. I don't see how to extend the polygon by only removing edges entirely visible from $p$. Because what's behind those edges could be arbitrarily convoluted. – Joseph O'Rourke Sep 22 '16 at 19:07

If the point $p$ is inside $P$, then "monotone with respect to a point" is called "star-shaped". If the point is outside, then I don't know of any special name.
An observation: if $p$ is outside, then a projective transformation sends a polygon/polyhedron monotone with respect to $p$ to one monotone with respect to a line. One has to send to infinity a line through $p$ that does not intersect $P$. That such a line exists follows from the monotonicity.
The answer to 2 is "no". A counterexample is provided by the Schoenhardt polyhedron. (Take $p$ above the top face.)