On the Reflexivity of Point Sets

@inproceedings{Arkin2001OnTR,
  title={On the Reflexivity of Point Sets},
  author={Esther M. Arkin and S{\'a}ndor P. Fekete and Ferran Hurtado and Joseph B. M. Mitchell and Marc Noy and Vera Sacrist{\'a}n Adinolfi and Saurabh Sethia},
  booktitle={WADS},
  year={2001}
}
We introduce a new measure for planar point sets S. Intuitively, it describes the combinatorial distance from a convex set: The reflexivity ρ(S) of S is given by the smallest number of reflex vertices in a simple polygonalization of S. We prove various combinatorial bounds and provide efficient algorithms to compute reflexivity, both exactly (in special cases) and approximately (in general). Our study naturally takes us into the examination of some closely related quantities, such as the convex… 

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References

SHOWING 1-10 OF 60 REFERENCES

On the convex layers of a planar set

  • B. Chazelle
  • Computer Science, Mathematics
    IEEE Trans. Inf. Theory
  • 1985
TLDR
An optimal algorithm is described for computing the convex layers of S, a set of n points in the Euclidean plane, for which optimal solutions are therefore known.

Fast Triangulation of the Plane with Respect to Simple Polygons

on a Partition into Convex Polygons

On Convex Decompositions of Points

TLDR
This paper shows that the minimum number of cells in a partition of a planar point set into convex cells, S, such that the union of the cells forms a simple polygon, P, and every point from S is on the boundary of P.

Partitioning point sets in space into disjoint convex polytopes

Searching for empty convex polygons

TLDR
It is shown that finding empty triangles is related to the problem of determining pairs of vertices that see each other in a star-shaped polygon, and a linear time algorithm for this problem which is of independent interest yields an optimal algorithm for finding all empty triangles.

Angle-Restricted Tours in the Plane

Enumerating Order Types for Small Point Sets with Applications

TLDR
A complete and reliable data base for all possible order types of size n=10 or less is established and substantiate the usefulness of the data base by applying it to several problems in computational and combinatorial geometry.

Enumerating order types for small sets with applications

TLDR
A complete and reliable data base for all possible order types of size $n=10$ or less is established and it is believed to be of value to many researchers who wish to examine their conjectures on small point configurations.
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