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.

Syntax

TR = triangulation(T,P)

TR = triangulation(T,x,y)

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

Description

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

T — Triangulation connectivity list

m-by-n matrix

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.

P — Points

matrix

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 — x-coordinates

column vector

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

y — y-coordinates

column vector

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

z — z-coordinates

column vector

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

Properties

Points — Triangulation points

matrix

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.

ConnectivityList — Triangulation connectivity list

matrix

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

2-D Triangulation

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