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Given a set of $n$ points in the Euclidean plane of which no three are collinear, does there always exist a convex triangulation and how can one be found algorithmically?

In this context a convex triangulation shall mean a triangulation in which the union of triangles with a common corner-point inside the convex hull constitute to a convex polygon.

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Nice idea but it seems not to always exist:


          ConvexTriangulation
          Point $4$'s star is reflex at $3$, Point $3$'s star is reflex at $4$.
Here's an argument that those are the only two triangulations. Each interior point ($3$ and $4$) must be degree-$3$ or degree-$4$: degree-$3$ to span more than $180^\circ$, and at most degree-$4$ because there are only four other points. In the left figure point $3$ has degree-$3$ and point $4$ degree-$4$, and in the right figure point $3$ has degree-$4$ and point $4$ degree-$3$. There must be a total of $9$ edges, $6$ of them diagonals. Then the diagonals in the two figures are forced.

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    $\begingroup$ It is an interesting question: When does a convex triangulation exist? And what is the computational complexity to decide if a set of points admits a convex triangulation? $\endgroup$ – Joseph O'Rourke Feb 22 at 0:02
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    $\begingroup$ Observation: If all neighbors of an interior vertex $v$ in a convex triangulation are interior vertices, then the degree of $v$ is at least $5$. $\endgroup$ – Jan Kyncl Feb 22 at 11:25
  • $\begingroup$ @JanKyncl: May I ask how you arrived at this interesting fact? $\endgroup$ – Joseph O'Rourke Feb 22 at 12:56
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    $\begingroup$ When looking at the figure, I noticed that around a vertex of degree at most $4$, two of the consecutive angles will sum up to more than 180 degrees. So the neighbor along the edge common to those two angles will have a nonconvex union of incident triangles. $\endgroup$ – Jan Kyncl Feb 22 at 14:18

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