# On the shape of a set of points in the plane

@inproceedings{Edelsbrunner1983OnTS, title={On the shape of a set of points in the plane}, author={Herbert Edelsbrunner and David G. Kirkpatrick and Raimund Seidel}, booktitle={IEEE Transactions on Information Theory}, year={1983} }

A generalization of the convex hull of a finite set of points in the plane is introduced and analyzed. This generalization leads to a family of straight-line graphs, " \alpha -shapes," which seem to capture the intuitive notions of "fine shape" and "crude shape" of point sets. It is shown that a-shapes are subgraphs of the closest point or furthest point Delaunay triangulation. Relying on this result an optimal O(n \log n) algorithm that constructs \alpha -shapes is developed.

## 1,546 Citations

### A-shapes of a finite point set

- MathematicsSCG '97
- 1997

A family of straightline graphs, derived from the Delaunay triangulation, is developed which is called A-shape of a finite point set (A is an arbitrary point set).

### Computing the shape of a point set in digital images

- Mathematics, Computer SciencePattern Recognit. Lett.
- 1993

### Iterative algorithm for computing the shape of a finite set of points

- Computer Science, MathematicsRemote Sensing
- 1994

An efficient algorithm for constructing the shape of a set of points is presented based on the relationship between the k-connected hulls sets an the Voronoi diagram.

### Efficient generation of simple polygons for characterizing the shape of a set of points in the plane

- Computer SciencePattern Recognit.
- 2008

### Quickly Placing a Point to Maximize Angles

- Computer Science, MathematicsCCCG
- 2014

This work seeks to place a new point q such that the constrained Delaunay triangulation of P[fqg has the largest possible minimum angle, and develops a simpler cubic-time algorithm quite dierent from the ones already known.

### Fast algorithm for computing the shape of a set of digital points

- Computer ScienceProceedings of 1st International Conference on Image Processing
- 1994

The true digital Voronoi tessellation in O(m/sup 2/) complexity is computed as a subgraph of the Delaunay triangulation as well as an image and the coordinates of the n initial points.

### An efficient incremental algorithm for generating the characteristic shape of a dynamic set of points in the plane

- Computer ScienceInt. J. Geogr. Inf. Sci.
- 2017

This paper presents for the first time an incremental χ-shape algorithm, capable of processing point data streams, that allows both insertion and deletion operations, and can handle streaming individual points and multiple point sets.

### Boundaries through Scattered Points of Unknown Density

- MathematicsCVGIP Graph. Model. Image Process.
- 1995

A method for the (re)construction of a simple closed polygon (2D) or polyhedron (3D) passing through all the points of a given set, based on a parameterized geometric graph, the γ-Neighborhood Graph.

### Recursive voids for identifying a nonconvex boundary of a set of points in the plane

- Computer Science, MathematicsPattern Recognit.
- 2013

### Dense point sets have sparse Delaunay triangulations

- Mathematics, Computer ScienceArXiv
- 2001

It is proved that the Delaunay triangulation of any set of n points in R^3 with spread D has complexity O(D^3), and this upper bound is tight in the worst case for all D = O(sqrt{n}.

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