# Documentation

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# voronoiDiagram

Voronoi diagram

## Syntax

```[V,R] = voronoiDiagram(DT) ```

## Description

```[V,R] = voronoiDiagram(DT)``` returns the Voronoi vertices, `V`, and the Voronoi regions, `R`, of the points, `DT.Points`.

The Voronoi diagram of a set of points, such as `DT.Points`, decomposes the space around each point, `DT.Points(j,:)`, into a region of influence, `R{j}`. Locations within the region, `R{j}`, are closer to point `j` than any other point in `DT.Points`. The region of influence is called the Voronoi region. The collection of all the Voronoi regions is the Voronoi diagram.

The Voronoi regions associated with points that lie on the convex hull of `DT.Points` are unbounded. Bounding edges of these regions radiate to infinity. The vertex at infinity is represented by the first vertex in `V`.

## Input Arguments

 `DT` A Delaunay triangulation, see `delaunayTriangulation`.

## Output Arguments

 `V` Voronoi vertices, returned as a matrix. Each row of `V` contains the coordinates of a Voronoi vertex. `R` Voronoi regions, returned as a vector cell array the same length as `DT.Points`. The elements of `R` are row numbers of `V`. The coordinates of the Voronoi vertices bounding a region are `V(R{j},:)`. The Voronoi region associated with the point `DT.Points(j)` is `R{j}`.

## Examples

expand all

Create a Delaunay triangulation from a set of points.

```P = [ 0.5 0 0 0.5 -0.5 -0.5 -0.2 -0.1 -0.1 0.1 0.1 -0.1 0.1 0.1 ]; DT = delaunayTriangulation(P);```

Calculate the Voronoi vertices and regions.

`[V,R] = voronoiDiagram(DT);`

Examine the connectivity of the Voronoi region associated with the third point in the triangulation.

`R{3}`
```ans = 1 10 7 4 ```

Examine the coordinates of the Voronoi vertices bounding the region.

`V(R{3},:)`
```ans = Inf Inf 0.7000 -1.6500 -0.0500 -0.5250 -1.7500 0.7500 ```

The `Inf` values indicate that the region contains points on the convex hull.