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# triangulation

Triangulation in 2-D or 3-D

## Description

Use triangulation to create an in-memory representation of any 2-D or 3-D triangulation data that is in matrix format, such as the matrix output from the delaunay function or other software tools. When your data is represented using triangulation, you can perform topological and geometric queries, which you can use to develop geometric algorithms. For example, you can find the triangles or tetrahedra attached to a vertex, those that share an edge, their circumcenters, and other features.

## Creation

To create a triangulation object, use the triangulation function with input arguments that define the triangulation's points and connectivity.

### Description

example

TR = triangulation(T,P) creates a 2-D or 3-D triangulation representation using the triangulation connectivity list T and the points in matrix P.

TR = triangulation(T,x,y) creates a 2-D triangulation representation with the point coordinates specified as column vectors x and y.

TR = triangulation(T,x,y,z) creates a 3-D triangulation representation with the point coordinates specified as column vectors x, y, and z.

### Input Arguments

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Triangulation connectivity list, specified as an m-by-n matrix, where m is the number of triangles or tetrahedra, and n is the number of vertices per triangle or tetrahedron. Each row of T contains the vertex IDs that define a triangle or tetrahedron. The vertex IDs are the row numbers of the input points. The ID of a triangle or tetrahedron in the triangulation is the corresponding row number in T.

Points, specified as a matrix whose columns are the x-coordinates, y-coordinates, and (possibly) z-coordinates of the triangulation points. The row numbers of P are the vertex IDs in the triangulation.

x-coordinates of triangulation points, specified as a column vector.

y-coordinates of triangulation points, specified as a column vector.

z-coordinates of triangulation points, specified as a column vector.

## Properties

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Triangulation points, represented as a matrix with the following characteristics:

• Each row in TR.Points contains the coordinates of a vertex.

• Each row number of TR.Points is a vertex ID.

Triangulation connectivity list, represented as a matrix with the following characteristics:

• Each element in TR.ConnectivityList is a vertex ID.

• Each row represents a triangle or tetrahedron in the triangulation.

• Each row number of TR.ConnectivityList is a triangle or tetrahedron ID.

## Object Functions

 barycentricToCartesian Convert coordinates from barycentric to Cartesian cartesianToBarycentric Convert coordinates from Cartesian to barycentric circumcenter Circumcenter of triangle or tetrahedron edgeAttachments Triangles or tetrahedra attached to specified edge edges Triangulation edges faceNormal Triangulation unit normal vectors featureEdges Sharp edges of surface triangulation freeBoundary Free boundary facets incenter Incenter of triangulation elements isConnected Test if two vertices are connected by an edge nearestNeighbor Vertex closest to specified point neighbors Triangle or tetrahedron neighbors pointLocation Triangle or tetrahedron enclosing point size Size of triangulation connectivity list vertexAttachments Triangles or tetrahedra attached to vertex vertexNormal Triangulation vertex normal

## Examples

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Define and plot the points in a 2-D triangulation.

P = [ 2.5    8.0
6.5    8.0
2.5    5.0
6.5    5.0
1.0    6.5
8.0    6.5];

Define the triangulation connectivity list.

T = [5  3  1;
3  2  1;
3  4  2;
4  6  2];

Create and plot the triangulation representation.

TR = triangulation(T,P)
TR =
triangulation with properties:

Points: [6x2 double]
ConnectivityList: [4x3 double]

triplot(TR)

Examine the coordinates of the vertices of the first triangle.

TR.Points(TR.ConnectivityList(1,:),:)
ans = 3×2

1.0000    6.5000
2.5000    5.0000
2.5000    8.0000

## Version History

Introduced in R2013a