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Triangulation in 2-D or 3-D


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.


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


TR = triangulation(T,P)
TR = triangulation(T,x,y)
TR = triangulation(T,x,y,z)



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 element in T is a vertex ID. Each row of T contains the vertex IDs that define a triangle or tetrahedron.

Points, specified as a matrix whose columns are the x-, y-, 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.


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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

barycentricToCartesianConvert point coordinates from barycentric to Cartesian
cartesianToBarycentricConvert point coordinates from Cartesian to barycentric
circumcenterCircumcenter of triangle or tetrahedron
edgeAttachmentsTriangles or tetrahedra attached to specified edge
edgesTriangulation edges
faceNormalTriangulation face normal
featureEdgesTriangulation sharp edges
freeBoundaryQuery free boundary facets
incenterIncenter of triangle or tetrahedron
isConnectedTest if two vertices are connected by edge
nearestNeighborVertex closest to specified location
neighborsNeighbors to specified triangle or tetrahedron
pointLocationTriangle or tetrahedron containing specified point
sizeSize of triangulation connectivity list
vertexAttachmentsTriangles or tetrahedra attached to specified vertex
vertexNormalTriangulation vertex normal


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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]


Examine the coordinates of the vertices of the first triangle.

ans = 

    1.0000    6.5000
    2.5000    5.0000
    2.5000    8.0000


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Introduced in R2013a

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